diff --git "a/P2P_nlin_CD.json" "b/P2P_nlin_CD.json" new file mode 100644--- /dev/null +++ "b/P2P_nlin_CD.json" @@ -0,0 +1,4002 @@ +[ + { + "text": "Exact Coherent Structures and Chaotic Dynamics in a Model of Cardiac\n Tissue: Unstable nonchaotic solutions embedded in the chaotic attractor can provide\nsignificant new insight into chaotic dynamics of both low- and high-dimensional\nsystems. In particular, in turbulent fluid flows, such unstable solutions are\nreferred to as exact coherent structures (ECS) and play an important role in\nboth initiating and sustaining turbulence. The nature of ECS and their role in\norganizing spatiotemporally chaotic dynamics, however, is reasonably well\nunderstood only for systems on relatively small spatial domains lacking\ncontinuous Euclidean symmetries. Construction of ECS on large domains and in\nthe presence of continuous translational and/or rotational symmetries remains a\nchallenge. This is especially true for models of excitable media which display\nspiral turbulence and for which the standard approach to computing ECS\ncompletely breaks down. This paper uses the Karma model of cardiac tissue to\nillustrate a potential approach that could allow computing a new class of ECS\non large domains of arbitrary shape by decomposing them into a patchwork of\nsolutions on smaller domains, or tiles, which retain Euclidean symmetries\nlocally.", + "category": "nlin_CD" + }, + { + "text": "A simple yet complex one-parameter family of generalized lorenz-like\n systems: This paper reports the finding of a simple one-parameter family of\nthree-dimensional quadratic autonomous chaotic systems. By tuning the only\nparameter, this system can continuously generate a variety of cascading\nLorenz-like attractors, which appears to be richer than the unified chaotic\nsystem that contains the Lorenz and the Chen systems as its two extremes.\nAlthough this new family of chaotic systems has very rich and complex dynamics,\nit has a very simple algebraic structure with only two quadratic terms (same as\nthe Lorenz and the Chen systems) and all nonzero coefficients in the linear\npart being -1 except one -0.1 (thus, simpler than the Lorenz and Chen systems).\nSurprisingly, although this new system belongs to the family of Lorenz-type\nsystems in some existing classifications such as the generalized Lorenz\ncanonical form, it can generate not only Lorenz-like attractors but also\nChen-like attractors. This suggests that there may exist some other unknown yet\nmore essential algebraic characteristics for classifying general\nthree-dimensional quadratic autonomous chaotic systems.", + "category": "nlin_CD" + }, + { + "text": "Polar rotation angle identifies elliptic islands in unsteady dynamical\n systems: We propose rotation inferred from the polar decomposition of the flow\ngradient as a diagnostic for elliptic (or vortex-type) invariant regions in\nnon-autonomous dynamical systems. We consider here two- and three-dimensional\nsystems, in which polar rotation can be characterized by a single angle. For\nthis polar rotation angle (PRA), we derive explicit formulas using the singular\nvalues and vectors of the flow gradient. We find that closed level sets of the\nPRA reveal elliptic islands in great detail, and singular level sets of the PRA\nuncover centers of such islands. Both features turn out to be objective\n(frame-invariant) for two-dimensional systems. We illustrate the diagnostic\npower of PRA for elliptic structures on several examples.", + "category": "nlin_CD" + }, + { + "text": "Stochastic Chaos and Predictability in Laboratory Earthquakes: Laboratory earthquakes exhibit characteristics of a low dimensional random\nattractor with a dimension similar to that of natural slow earthquakes. A model\nof stochastic differential equations based on rate and state-dependent friction\nexplains the laboratory observations. We study the transition from stable\nsliding to stickslip events and find that aperiodic behavior can be explained\nby small perturbations in the stress state. Friction's nonlinear nature\namplifies small scale perturbations, reducing the predictability of the\notherwise periodic macroscopic dynamics.", + "category": "nlin_CD" + }, + { + "text": "On the non-randomness of maximum Lempel Ziv complexity sequences of\n finite size: Random sequences attain the highest entropy rate. The estimation of entropy\nrate for an ergodic source can be done using the Lempel Ziv complexity measure\nyet, the exact entropy rate value is only reached in the infinite limit. We\nprove that typical random sequences of finite length fall short of the maximum\nLempel-Ziv complexity, contrary to common belief. We discuss that, for a finite\nlength, maximum Lempel-Ziv sequences can be built from a well defined\ngenerating algorithm, which makes them of low Kolmogorov-Chaitin complexity,\nquite the opposite to randomness. It will be discussed that Lempel-Ziv measure\nis, in this sense, less general than Kolmogorov-Chaitin complexity, as it can\nbe fooled by an intelligent enough agent. The latter will be shown to be the\ncase for the binary expansion of certain irrational numbers. Maximum Lempel-Ziv\nsequences induce a normalization that gives good estimates of entropy rate for\nseveral sources, while keeping bounded values for all sequence length, making\nit an alternative to other normalization schemes in use.", + "category": "nlin_CD" + }, + { + "text": "Understanding deterministic diffusion by correlated random walks: Low-dimensional periodic arrays of scatterers with a moving point particle\nare ideal models for studying deterministic diffusion. For such systems the\ndiffusion coefficient is typically an irregular function under variation of a\ncontrol parameter. Here we propose a systematic scheme of how to approximate\ndeterministic diffusion coefficients of this kind in terms of correlated random\nwalks. We apply this approach to two simple examples which are a\none-dimensional map on the line and the periodic Lorentz gas. Starting from\nsuitable Green-Kubo formulas we evaluate hierarchies of approximations for\ntheir parameter-dependent diffusion coefficients. These approximations converge\nexactly yielding a straightforward interpretation of the structure of these\nirregular diffusion coeficients in terms of dynamical correlations.", + "category": "nlin_CD" + }, + { + "text": "Navier-Stokes-alpha model: LES equations with nonlinear dispersion: We present a framework for discussing LES equations with nonlinear\ndispersion. In this framework, we discuss the properties of the nonlinearly\ndispersive Navier-Stokes-alpha model of incompressible fluid turbulence ---\nalso called the viscous Camassa-Holm equations and the LANS equations in the\nliterature --- in comparison with the corresponding properties of large eddy\nsimulation (LES) equations obtained via the approximate-inverse approach.\n In this comparison, we identify the spatially filtered NS-alpha equations\nwith a class of generalized LES similarity models. Applying a certain\napproximate inverse to this class of LES models restores the Kelvin circulation\ntheorem for the defiltered velocity and shows that the NS-alpha model describes\nthe dynamics of the defiltered velocity for this class of generalized LES\nsimilarity models. We also show that the subgrid scale forces in the NS-alpha\nmodel transform covariantly under Galilean transformations and under a change\nto a uniformly rotating reference frame. Finally, we discuss in the spectral\nformulation how the NS-alpha model retains the local interactions among the\nlarge scales, retains the nonlocal sweeping effects of large scales on small\nscales, yet attenuates the local interactions of the small scales amongst\nthemselves.", + "category": "nlin_CD" + }, + { + "text": "Uniform framework for the recurrence-network analysis of chaotic time\n series: We propose a general method for the construction and analysis of unweighted\n$\\epsilon$ - recurrence networks from chaotic time series. The selection of the\ncritical threshold $\\epsilon_c$ in our scheme is done empirically and we show\nthat its value is closely linked to the embedding dimension $M$. In fact, we\nare able to identify a small critical range $\\Delta \\epsilon$ numerically that\nis approximately the same for the random and several standard chaotic time\nseries for a fixed $M$. This provides us a uniform framework for the non\nsubjective comparison of the statistical measures of the recurrence networks\nconstructed from various chaotic attractors. We explicitly show that the degree\ndistribution of the recurrence network constructed by our scheme is\ncharacteristic to the structure of the attractor and display statistical scale\ninvariance with respect to increase in the number of nodes $N$. We also present\ntwo practical applications of the scheme, detection of transition between two\ndynamical regimes in a time delayed system and identification of the\ndimensionality of the underlying system from real world data with limited\nnumber of points, through recurrence network measures. The merits, limitations\nand the potential applications of the proposed method have also been\nhighlighted.", + "category": "nlin_CD" + }, + { + "text": "Nonlinear dynamics of molecular superrotors: We consider a diatomic molecule driven by a linearly polarized laser pulse\nwith a polarization axis rotating with a constant acceleration. This setup is\nreferred to as optical centrifuge, and it is known to lead to high-angular\nmomenta for the molecule (superrotor states) and, possibly, to dissociation.\nHere we elucidate the dynamical mechanisms behind the creation of superrotor\nstates and their dissociation. We unravel the role of the various parameters of\nthe laser field in these processes by considering reduced Hamiltonian models\nencapsulating the different phases in the creation of superrotor states,\npossibly leading to dissociation.", + "category": "nlin_CD" + }, + { + "text": "Statistics of branched flow in a weak correlated random potential: Recent images of electron flow through a two-dimensional electron gas (2DEG)\ndevice show branching behavior that is reproduced in numerical simulations of\nmotion in a correlated random potential [cond-mat/0010348]. We show how such\nbranching naturally arises from caustics in the classical flow and find a\nsimple scaling behavior of the branching under variation of the random\npotential strength. Analytic results describing the statistical properties of\nthe branching are confirmed by classical and quantum numerical tests.", + "category": "nlin_CD" + }, + { + "text": "Complexity in atoms: an approach with a new analytical density: In this work, the calculation of complexity on atomic systems is considered.\nIn order to unveil the increasing of this statistical magnitude with the atomic\nnumber due to the relativistic effects, recently reported in [A. Borgoo, F. De\nProft, P. Geerlings, K.D. Sen, Chem. Phys. Lett., 444 (2007) 186], a new\nanalytical density to describe neutral atoms is proposed. This density is\ninspired in the Tietz potential model. The parameters of this density are\ndetermined from the normalization condition and from a variational calculation\nof the energy, which is a functional of the density. The density is\nnon-singular at the origin and its specific form is selected so as to fit the\nresults coming from non-relativistic Hartree-Fock calculations. The main\ningredients of the energy functional are the non-relativistic kinetic energy,\nthe nuclear-electron attraction energy and the classical term of the electron\nrepulsion. The relativistic correction to the kinetic energy and the Weizsacker\nterm are also taken into account. The Dirac and the correlation terms are shown\nto be less important than the other terms and they have been discarded in this\nstudy. When the statistical measure of complexity is calculated in position\nspace with the analytical density derived from this model, the increasing trend\nof this magnitude as the atomic number increases is also found.", + "category": "nlin_CD" + }, + { + "text": "Chaotic Spin Dynamics of a Long Nanomagnet Driven by a Current: We study the spin dynamics of a long nanomagnet driven by an electrical\ncurrent. In the case of only DC current, the spin dynamics has a sophisticated\nbifurcation diagram of attractors. One type of attractors is a weak chaos. On\nthe other hand, in the case of only AC current, the spin dynamics has a rather\nsimple bifurcation diagram of attractors. That is, for small Gilbert damping,\nwhen the AC current is below a critical value, the attractor is a limit cycle;\nabove the critical value, the attractor is chaotic (turbulent). For normal\nGilbert damping, the attractor is always a limit cycle in the physically\ninteresting range of the AC current. We also developed a Melnikov integral\ntheory for a theoretical prediction on the occurrence of chaos. Our Melnikov\nprediction seems performing quite well in the DC case. In the AC case, our\nMelnikov prediction seems predicting transient chaos. The sustained chaotic\nattractor seems to have extra support from parametric resonance leading to a\nturbulent state.", + "category": "nlin_CD" + }, + { + "text": "On spurious detection of linear response and misuse of the\n fluctuation-dissipation theorem in finite time series: Using a sensitive statistical test we determine whether or not one can detect\nthe breakdown of linear response given observations of deterministic dynamical\nsystems. A goodness-of-fit statistics is developed for a linear statistical\nmodel of the observations, based on results on central limit theorems for\ndeterministic dynamical systems, and used to detect linear response breakdown.\nWe apply the method to discrete maps which do not obey linear response and show\nthat the successful detection of breakdown depends on the length of the time\nseries, the magnitude of the perturbation and on the choice of the observable.\n We find that in order to reliably reject the assumption of linear response\nfor typical observables sufficiently large data sets are needed. Even for\nsimple systems such as the logistic map, one needs of the order of $10^6$\nobservations to reliably detect the breakdown, if less observations are\navailable one may be falsely led to conclude that linear response theory is\nvalid. The amount of data required is larger the smaller the applied\nperturbation. For judiciously chosen observables the necessary amount of data\ncan be drastically reduced, but requires detailed {\\em{a priori}} knowledge\nabout the invariant measure which is typically not available for complex\ndynamical systems.\n Furthermore we explore the use of the fluctuation-dissipation theorem (FDT)\nin cases with limited data length or coarse-graining of observations. The FDT,\nif applied naively to a system without linear response, is shown to be very\nsensitive to the details of the sampling method, resulting in erroneous\npredictions of the response.", + "category": "nlin_CD" + }, + { + "text": "Survival Probability for the Stadium Billiard: We consider the open stadium billiard, consisting of two semicircles joined\nby parallel straight sides with one hole situated somewhere on one of the\nsides. Due to the hyperbolic nature of the stadium billiard, the initial decay\nof trajectories, due to loss through the hole, appears exponential. However,\nsome trajectories (bouncing ball orbits) persist and survive for long times and\ntherefore form the main contribution to the survival probability function at\nlong times. Using both numerical and analytical methods, we concur with\nprevious studies that the long-time survival probability for a reasonably small\nhole drops like Constant/time; here we obtain an explicit expression for the\nConstant.", + "category": "nlin_CD" + }, + { + "text": "Convergence of direct recursive algorithm for identification of Preisach\n hysteresis model with stochastic input: We consider a recursive iterative algorithm for identification of parameters\nof the Preisach model, one of the most commonly used models of hysteretic\ninput-output relationships. The classical identification algorithm due to\nMayergoyz defines explicitly a series of test inputs that allow one to find\nparameters of the Preisach model with any desired precision provided that (a)\nsuch input time series can be implemented and applied; and, (b) the\ncorresponding output data can be accurately measured and recorded. Recursive\niterative identification schemes suitable for a number of engineering\napplications have been recently proposed as an alternative to the classical\nalgorithm. These recursive schemes do not use any input design but rather rely\non an input-output data stream resulting from random fluctuations of the input\nvariable. Furthermore, only recent values of the input-output data streams are\navailable for the scheme at any time instant. In this work, we prove\nexponential convergence of such algorithms, estimate explicitly the convergence\nrate, and explore which properties of the stochastic input and the algorithm\naffect the guaranteed convergence rate.", + "category": "nlin_CD" + }, + { + "text": "A novel type of spiral wave with trapped ions: Pattern formation in ultra-cold quantum systems has recently received a great\ndeal of attention.In this work, we investigate a two-dimensional model system\naccounting for the dynamics of trapped ions. We find a novel spiral wave which\nis rigidly rotating but with a peculiar core region in which adjacent ions\noscillate in anti-phase. The formation of this novel spiral wave is ascribed to\nthe novel excitability reported by Lee and Cross. The breakup of the novel\nspiral wave is probed and, especially, one extraordinary scenario of the\ndisappearance of spiral wave caused by spontaneous expansion of the anti-phased\ncore is unveiled.", + "category": "nlin_CD" + }, + { + "text": "Exponential Recovery of Low Frequency Fluctuations in a Diode Laser with\n Optical Feedback: We show that the recovery after each power drop on the chaotic Low Frequency\nFluctuations in a semiconductor laser with optical feedback follows an\nexponential envelope. The time constant for such exponential behavior was\nexperimentally measured. This recovery time constant and the average time\ninterval between consecutive drops are shown to have different dependences when\nmeasured as function of the pump current.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of a Charged Thomas Oscillator in an External Magnetic Field: In this letter, we provide a detailed numerical examination of the dynamics\nof a charged Thomas oscillator in an external magnetic field. We do so by\nadopting and then modifying the cyclically symmetric Thomas oscillator to study\nthe dynamics of a charged particle in an external magnetic field. These\ndynamical behaviours for weak and strong field strength parameters fall under\ntwo categories; conservative and dissipative. The system shows a complex\nquasi-periodic attractor whose topology depends on initial conditions for high\nfield strengths in the conservative regime. There is a transition from\nadiabatic motion to chaos on decreasing the field strength parameter. In the\ndissipative regime, the system is chaotic for weak field strength and weak\ndamping but shows a limit cycle for high field strengths. Such behaviour is due\nto an additional negative feedback loop that comes into action at high field\nstrengths and forces the system dynamics to be stable in periodic oscillations.\nFor weak damping and weak field strength, the system dynamics mimic Brownian\nmotion via chaotic walks.", + "category": "nlin_CD" + }, + { + "text": "Anomalous shell effect in the transition from a circular to a triangular\n billiard: We apply periodic orbit theory to a two-dimensional non-integrable billiard\nsystem whose boundary is varied smoothly from a circular to an equilateral\ntriangular shape. Although the classical dynamics becomes chaotic with\nincreasing triangular deformation, it exhibits an astonishingly pronounced\nshell effect on its way through the shape transition. A semiclassical analysis\nreveals that this shell effect emerges from a codimension-two bifurcation of\nthe triangular periodic orbit. Gutzwiller's semiclassical trace formula, using\na global uniform approximation for the bifurcation of the triangular orbit and\nincluding the contributions of the other isolated orbits, describes very well\nthe coarse-grained quantum-mechanical level density of this system. We also\ndiscuss the role of discrete symmetry for the large shell effect obtained here.", + "category": "nlin_CD" + }, + { + "text": "Noise-enhanced chaos in a weakly coupled GaAs/(Al,Ga)As superlattice: Noise-enhanced chaos in a doped, weakly coupled GaAs/Al_{0.45}Ga_{0.55}As\nsuperlattice has been observed at room temperature in experiments as well as in\nthe results of the simulation of nonlinear transport based on a discrete\ntunneling model. When external noise is added, both the measured and simulated\ncurrent-versus-time traces contain irregularly spaced spikes for particular\napplied voltages, which separate a regime of periodic current oscillations from\na region of no current oscillations at all. In the voltage region without\ncurrent oscillations, the electric field profile consist of a low-field domain\nnear the emitter contact separated by a domain wall consisting of a charge\naccumulation layer from a high-field regime closer to the collector contact.\nWith increasing noise amplitude, spontaneous chaotic current oscillations\nappear over a wider bias voltage range. For these bias voltages, the domain\nboundary between the two electric-field domains becomes unstable, and very\nsmall current or voltage fluctuations can trigger the domain boundary to move\ntoward the collector and induce chaotic current spikes. The experimentally\nobserved features are qualitatively very well reproduced by the simulations.\nIncreased noise can consequently enhance chaotic current oscillations in\nsemiconductor superlattices.", + "category": "nlin_CD" + }, + { + "text": "Dynamical Taxonomy: some taxonomic ranks to systematically classify\n every chaotic attractor: Characterizing accurately chaotic behaviors is not a trivial problem and must\nallow to determine the properties that two given chaotic invariant sets share\nor not. The underlying problem is the classification of chaotic regimes, and\ntheir labelling. Addressing these problems correspond to the development of a\ndynamical taxonomy, exhibiting the key properties discriminating the variety of\nchaotic behaviors discussed in the abundant literature. Starting from the\nhierarchy of chaos initially proposed by one of us, we systematized the\ndescription of chaotic regimes observed in three- and four-dimensional spaces,\nwhich cover a large variety of known (and less known) examples of chaotic\nsystems. Starting with the spectrum of Lyapunov exponents as the first\ntaxonomic ranks, we extended the description to higher ranks with some concepts\ninherited from topology (bounding torus, surface of section, first-return\nmap...).\n By treating extensively the R\\\"ossler and the Lorenz attractors, we extended\nthe description of branched manifold -- the highest taxonomic rank for\nclassifying chaotic attractor -- by a linking matrix (or linker) to\nmulti-component attractors (bounded by a torus whose genus g <= 3", + "category": "nlin_CD" + }, + { + "text": "Artin Billiard Exponential Decay of Correlation Functions: The hyperbolic Anosov C-systems have exponential instability of their\ntrajectories and as such represent the most natural chaotic dynamical systems.\nOf special interest are C-systems which are defined on compact surfaces of the\nLobachevsky plane of constant negative curvature. An example of such system has\nbeen introduced in a brilliant article published in 1924 by the mathematician\nEmil Artin. The dynamical system is defined on the fundamental region of the\nLobachevsky plane which is obtained by the identification of points congruent\nwith respect to the modular group, a discrete subgroup of the Lobachevsky plane\nisometries. The fundamental region in this case is a hyperbolic triangle. The\ngeodesic trajectories of the non-Euclidean billiard are bounded to propagate on\nthe fundamental hyperbolic triangle. In this article we shall expose his\nresults, will calculate the correlation functions/observables which are defined\non the phase space of the Artin billiard and demonstrate the exponential decay\nof the correlation functions with time. We use Artin symbolic dynamics, the\ndifferential geometry and group theoretical methods of Gelfand and Fomin.", + "category": "nlin_CD" + }, + { + "text": "Extreme Wave Events in Directional, Random Oceanic Sea States: We discuss the effects of the directional spreading on the occurrence of\nextreme wave events. We numerically integrate the envelope equation recently\nproposed by Trulsen et al., Phys of Fluids 2000, as a weakly nonlinear model\nfor realistic oceanic gravity waves.Initial conditions for numerical\nsimulations are characterized by the spatial JONSWAP power spectrum for several\nvalues of the significant wave height, steepness and directional spreading. We\nshow that by increasing the directionality of the initial spectrum the\nappearance of extreme events is notably reduced.", + "category": "nlin_CD" + }, + { + "text": "3D chaotic model for sub-grid turbulent dispersion in Large Eddy\n Simulations: We introduce a 3D multiscale kinematic velocity field as a model to simulate\nLagrangian turbulent dispersion. The incompressible velocity field is a\nnonlinear deterministic function, periodic in space and time, that generates\nchaotic mixing of Lagrangian trajectories. Relative dispersion properties, e.g.\nthe Richardson's law, are correctly reproduced under two basic conditions: 1)\nthe velocity amplitudes of the spatial modes must be related to the\ncorresponding wavelengths through the Kolmogorov scaling; 2) the problem of the\nlack of \"sweeping effect\" of the small eddies by the large eddies, common to\nkinematic simulations, has to be taken into account. We show that, as far as\nLagrangian dispersion is concerned, our model can be successfully applied as\nadditional sub-grid contribution for Large Eddy Simulations of the planetary\nboundary layer flow.", + "category": "nlin_CD" + }, + { + "text": "Effect of pitchfork bifurcations on the spectral statistics of\n Hamiltonian systems: We present a quantitative semiclassical treatment of the effects of\nbifurcations on the spectral rigidity and the spectral form factor of a\nHamiltonian quantum system defined by two coupled quartic oscillators, which on\nthe classical level exhibits mixed phase space dynamics. We show that the\nsignature of a pitchfork bifurcation is two-fold: Beside the known effect of an\nenhanced periodic orbit contribution due to its peculiar $\\hbar$-dependence at\nthe bifurcation, we demonstrate that the orbit pair born {\\em at} the\nbifurcation gives rise to distinct deviations from universality slightly {\\em\nabove} the bifurcation. This requires a semiclassical treatment beyond the\nso-called diagonal approximation. Our semiclassical predictions for both the\ncoarse-grained density of states and the spectral rigidity, are in excellent\nagreement with corresponding quantum-mechanical results.", + "category": "nlin_CD" + }, + { + "text": "Soft wall effects on interacting particles in billiards: The effect of physically realizable wall potentials (soft walls) on the\ndynamics of two interacting particles in a one-dimensional (1D) billiard is\nexamined numerically. The 1D walls are modeled by the error function and the\ntransition from hard to soft walls can be analyzed continuously by varying the\nsoftness parameter $\\sigma$. For $\\sigma\\to 0$ the 1D hard wall limit is\nobtained and the corresponding wall force on the particles is the\n$\\delta$-function. In this limit the interacting particle dynamics agrees with\nprevious results obtained for the 1D hard walls. We show that the two\ninteracting particles in the 1D soft walls model is equivalent to one particle\ninside a soft right triangular billiard. Very small values of $\\sigma$\nsubstantiously change the dynamics inside the billiard and the mean finite-time\nLyapunov exponent decreases significantly as the consequence of regular islands\nwhich appear due to the low-energy double collisions (simultaneous\nparticle-particle-1D wall collisions). The rise of regular islands and sticky\ntrajectories induced by the 1D wall softness is quantified by the number of\noccurrences of the most probable finite-time Lyapunov exponent. On the other\nhand, chaotic motion in the system appears due to the high-energy double\ncollisions. In general we observe that the mean finite-time Lyapunov exponent\ndecreases when $\\sigma$ increases, but the number of occurrences of the most\nprobable finite-time Lyapunov exponent increases, meaning that the phase-space\ndynamics tends to be more ergodiclike. Our results suggest that the transport\nefficiency of interacting particles and heat conduction in periodic structures\nmodeled by billiards will strongly be affected by the smoothness of physically\nrealizable walls.", + "category": "nlin_CD" + }, + { + "text": "The permutation entropy rate equals the metric entropy rate for ergodic\n information sources and ergodic dynamical systems: Permutation entropy quantifies the diversity of possible orderings of the\nvalues a random or deterministic system can take, as Shannon entropy quantifies\nthe diversity of values. We show that the metric and permutation entropy\nrates--measures of new disorder per new observed value--are equal for ergodic\nfinite-alphabet information sources (discrete-time stationary stochastic\nprocesses). With this result, we then prove that the same holds for\ndeterministic dynamical systems defined by ergodic maps on $n$% -dimensional\nintervals. This result generalizes a previous one for piecewise monotone\ninterval maps on the real line (Bandt, Keller and Pompe, \"Entropy of interval\nmaps via permutations\",\\textit{Nonlinearity} \\textbf{15}, 1595-602, (2002)), at\nthe expense of requiring ergodicity and using a definition of permutation\nentropy rate differing in the order of two limits. The case of non-ergodic\nfinite-alphabet sources is also studied and an inequality developed. Finally,\nthe equality of permutation and metric entropy rates is extended to ergodic\nnon-discrete information sources when entropy is replaced by differential\nentropy in the usual way.", + "category": "nlin_CD" + }, + { + "text": "Synchronization of nearly-identical dynamical systems. II Optimized\n networks: In this paper we use the master stability function (MSF) for nearly identical\ndynamical systems obtained in the previous paper to construct optimized\nnetworks (ONs) which show better synchronizability. Nearly identical nature is\nthe result of having some node dependent parameters (NDPs) in the dynamics. We\nstudy the correlation between various network properties and the values of NDPs\non different nodes for the optimized networks and compare them with random\nnetworks using the example of coupled R\\\"ossler systems. In an ON, the nodes\nwith NDP values at one extreme, e.g. nodes with higher frequencies in coupled\nR\\\"ossler systems, have higher degrees and are chosen as hubs. These nodes also\nshow higher betweenness centrality. The links in ON are preferably between\nnodes with large differences in NDP values. The ONs have in general higher\nclustering coefficient. We also study other network properties such as average\nshortest path, degree mixing etc. and their relation to the NDP in ON. We\nconsider cases of both one and two NDPs and also directed networks.", + "category": "nlin_CD" + }, + { + "text": "Characterizing Weak Chaos using Time Series of Lyapunov Exponents: We investigate chaos in mixed-phase-space Hamiltonian systems using time\nseries of the finite- time Lyapunov exponents. The methodology we propose uses\nthe number of Lyapunov exponents close to zero to define regimes of ordered\n(stickiness), semi-ordered (or semi-chaotic), and strongly chaotic motion. The\ndynamics is then investigated looking at the consecutive time spent in each\nregime, the transition between different regimes, and the regions in the\nphase-space associated to them. Applying our methodology to a chain of coupled\nstandard maps we obtain: (i) that it allows for an improved numerical\ncharacterization of stickiness in high-dimensional Hamiltonian systems, when\ncompared to the previous analyses based on the distribution of recurrence\ntimes; (ii) that the transition probabilities between different regimes are\ndetermined by the phase-space volume associated to the corresponding regions;\n(iii) the dependence of the Lyapunov exponents with the coupling strength.", + "category": "nlin_CD" + }, + { + "text": "Synchronizing spatio-temporal chaos with imperfect models: a stochastic\n surface growth picture: We study the synchronization of two spatially extended dynamical systems\nwhere the models have imperfections. We show that the synchronization error\nacross space can be visualized as a rough surface governed by the\nKardar-Parisi-Zhang equation with both upper and lower bounding walls\ncorresponding to nonlinearities and model discrepancies, respectively. Two\ntypes of model imperfections are considered: parameter mismatch and unresolved\nfast scales, finding in both cases the same qualitative results. The\nconsistency between different setups and systems indicates the results are\ngeneric for a wide family of spatially extended systems.", + "category": "nlin_CD" + }, + { + "text": "Loschmidt echo in quantum maps: the elusive nature of the Lyapunov\n regime: The Loschmidt echo is a measure of the stability and reversibility of quantum\nevolution under perturbations of the Hamiltonian. One of the expected and most\nrelevant characteristics of this quantity for chaotic systems is an exponential\ndecay with a perturbation independent decay rate given by the classical\nLyapunov exponent. However, a non-uniform decay -- instead of the Lyapunov\nregime -- has been reported in several systems. In this work we find an\nanalytical semiclassical expression for the averaged fidelity amplitude that\ncan be related directly to the anomalous -- unexpected-- behaviour of the LE.", + "category": "nlin_CD" + }, + { + "text": "Nonquasilinear evolution of particle velocity in incoherent waves with\n random amplitudes: The one-dimensional motion of $N$ particles in the field of many incoherent\nwaves is revisited numerically. When the wave complex amplitudes are\nindependent, with a gaussian distribution, the quasilinear approximation is\nfound to always overestimate transport and to become accurate in the limit of\ninfinite resonance overlap.", + "category": "nlin_CD" + }, + { + "text": "An Introduction to Quantum Chaos: Nonlinear dynamics (``chaos theory'') and quantum mechanics are two of the\nscientific triumphs of the 20th century. The former lies at the heart of the\nmodern interdisciplinary approach to science, whereas the latter has\nrevolutionized physics. Both chaos theory and quantum mechanics have achieved a\nfairly large level of glamour in the eyes of the general public. The study of\nquantum chaos encompasses the application of dynamical systems theory in the\nquantum regime. In the present article, we give a brief review of the origin\nand fundamentals of both quantum mechanics and nonlinear dynamics. We recount\nthe birth of dynamical systems theory and contrast chaotic motion with\nintegrable motion. We similarly recall the transition from classical to quantum\nmechanics and discuss the origin of the latter. We then consider the interplay\nbetween nonlinear dynamics and quantum mechanics via a classification and\nexplanation of the three types of quantum chaos. We include several recent\nresults in this discussion.", + "category": "nlin_CD" + }, + { + "text": "Growth and performance of the periodic orbits of a nonlinear driven\n oscillator: Periodic orbits are fundamental to understand the dynamics of nonlinear\nsystems. In this work, we focus on two aspects of interest regarding periodic\norbits, in the context of a dissipative mapping, derived from a prototype model\nof a non-linear driven oscillator with fast relaxation and a limit cycle. For\nthis map, we show numerically the exponential growth of periodic orbits\nquantity in certain regions of the parameter space and provide an analytical\nbound for such growth rate, by making use of the transition matrix associated\nwith a given periodic orbit. Furthermore, we give numerical evidence to support\nthat optimal orbits, those that maximize time averages, are often unstable\nperiodic orbits with low period, by numerically comparing their performance\nunder a family of sinusoidal functions.", + "category": "nlin_CD" + }, + { + "text": "Markovian solutions of inviscid Burgers equation: For solutions of (inviscid, forceless, one dimensional) Burgers equation with\nrandom initial condition, it is heuristically shown that a stationary\nFeller-Markov property (with respect to the space variable) at some time is\nconserved at later times, and an evolution equation is derived for the\ninfinitesimal generator. Previously known explicit solutions such as\nFrachebourg-Martin's (white noise initial velocity) and Carraro-Duchon's L\\'evy\nprocess intrinsic-statistical solutions (including Brownian initial velocity)\nare recovered as special cases.", + "category": "nlin_CD" + }, + { + "text": "The Limit Cycles of Lienard Equations in the Strongly Non-Linear Regime: Lienard systems of the form $\\ddot{x}+\\epsilon f(x)\\dot{x}+x=0$, with f(x) an\neven function, are studied in the strongly nonlinear regime\n($\\epsilon\\to\\infty$). A method for obtaining the number, amplitude and loci of\nthe limit cycles of these equations is derived. The accuracy of this method is\nchecked in several examples. Lins-Melo-Pugh conjecture for the polynomial case\nis true in this regime.", + "category": "nlin_CD" + }, + { + "text": "Isentropic perturbations of a chaotic domain: Three major properties of the chaotic dynamics of the standard map, namely,\nthe measure \\mu of the main connected chaotic domain, the maximum Lyapunov\nexponent L of the motion in this domain, and the dynamical entropy h = \\mu L\nare studied as functions of the stochasticity parameter K. The perturbations of\nthe domain due to emergence and disintegration of islands of stability, upon\nsmall variations of K, are considered in particular. By means of extensive\nnumerical experiments, it is shown that these perturbations are isentropic (at\nleast approximately). In other words, the dynamical entropy does not fluctuate,\nwhile local jumps in \\mu and L are significant.", + "category": "nlin_CD" + }, + { + "text": "Paul Trap and the Problem of Quantum Stability: This work is devoted to the investigation of possibility of controlling of\nions motion inside Paul trap. It has been shown that by proper selection of the\nparameters of controlling electric fields, stable localization of ions inside\nPaul trap is possible. Quantum consideration of this problem is reduced to the\ninvestigation of the Mathieu-Schrodinger equation. It has been shown that\nquantum consideration is appreciably different from classical one that leads to\nstronger limitations of the values of the parameters of stable motion.\nConnection between the problem under study and the possibility of experimental\nobservation of quantum chaos has been shown.", + "category": "nlin_CD" + }, + { + "text": "Phase Space Analysis of the Non-Existence of Dynamical Matching in a\n Stretched Caldera Potential Energy Surface: In this paper we continue our studies of the two dimensional caldera\npotential energy surface in a parametrized family that allows for a study of\nthe effect of symmetry on the phase space structures that govern how\ntrajectories enter, cross, and exit the region of the caldera. As a particular\nform of trajectory crossing, we are able to determine the effect of symmetry\nand phase space structure on dynamical matching. We show that there is a\ncritical value of the symmetry parameter which controls the phase space\nstructures responsible for the manner of crossing, interacting with the central\nregion (including trapping in this region) and exiting the caldera. We provide\nan explanation for the existence of this critical value in terms of the\nbehavior of the Henon stability parameter for the associated periodic orbits.", + "category": "nlin_CD" + }, + { + "text": "Transport in Thin Gravity-driven Flow over a Curved Substrate: We consider steady gravity-driven flow of a thin layer of viscous fluid over\na curved substrate. The substrate has topographical variations (`bumps') on a\nlarge scale compared to the layer thickness. Using lubrication theory, we find\nthe velocity field in generalized curvilinear coordinates. We correct the\nvelocity field so as to satisfy kinematic constraints, which is essential to\navoid particles escaping the fluid when computing their trajectories. We then\ninvestigate the particle transport properties of flows over substrates with\ntranslational symmetry, where chaotic motion is precluded. The existence of\ntrapped and untrapped trajectories leads to complicated transport properties\neven for this simple case. For more general substrate shapes, the trajectories\nchaotically jump between trapped and untrapped motions.", + "category": "nlin_CD" + }, + { + "text": "Ordered level spacing probability densities: Spectral statistics of quantum systems have been studied in detail using the\nnearest neighbour level spacings, which for generic chaotic systems follows\nrandom matrix theory predictions. In this work, the probability density of the\nclosest neighbour and farther neighbour spacings from a given level are\nintroduced. Analytical predictions are derived using a $3 \\times 3$ matrix\nmodel. The closest neighbour density is generalized to the $k-$th closest\nneighbour spacing density, which allows for investigating long-range\ncorrelations. For larger $k$ the probability density of $k-$th closest\nneighbour spacings is well described by a Gaussian. Using these $k-$th closest\nneighbour spacings we propose the ratio of the closest neighbour to the second\nclosest neighbour as an alternative to the ratio of successive spacings. For a\nPoissonian spectrum the density of the ratio is flat, whereas for the three\nGaussian ensembles repulsion at small values is found. The ordered spacing\nstatistics and their ratio are numerically studied for the integrable circle\nbilliard, the chaotic cardioid billiard, the standard map and the zeroes of the\nRiemann zeta function. Very good agreement with the predictions is found.", + "category": "nlin_CD" + }, + { + "text": "Hidden chaotic attractors and chaos suppression in an impulsive discrete\n economical supply and demand dynamical system: Impulsive control is used to suppress the chaotic behavior in an\none-dimensional discrete supply and demand dynamical system. By perturbing\nperiodically the state variable with constant impulses, the chaos can be\nsuppressed. It is proved analytically that the obtained orbits are bounded and\nperiodic. Moreover, it is shown for the first time that the difference\nequations with impulses, used to control the chaos, can generate hidden chaotic\nattractors. To the best of the authors knowledge, this interesting feature has\nnot yet been discussed. The impulsive algorithm can be used to stabilize chaos\nin other classes of discrete dynamical systems.", + "category": "nlin_CD" + }, + { + "text": "Chaotic Geodesics: When a shallow layer of inviscid fluid flows over a substrate, the fluid\nparticle trajectories are, to leading order in the layer thickness, geodesics\non the two-dimensional curved space of the substrate. Since the two-dimensional\ngeodesic equation is a two degree-of-freedom autonomous Hamiltonian system, it\ncan exhibit chaos, depending on the shape of the substrate. We find chaotic\nbehaviour for a range of substrates.", + "category": "nlin_CD" + }, + { + "text": "Quantum chaos and its kinetic stage of evolution: Usually reason of irreversibility in open quantum-mechanical system is\ninteraction with a thermal bath, consisting form infinite number of degrees of\nfreedom. Irreversibility in the system appears due to the averaging over all\npossible realizations of the environment states. But, in case of open\nquantum-mechanical system with few degrees of freedom situation is much more\ncomplicated. Should one still expect irreversibility, if external perturbation\nis just an adiabatic force without any random features? Problem is not clear\nyet. This is main question we address in this review paper. We prove that key\npoint in the formation of irreversibility in chaotic quantum-mechanical systems\nwith few degrees of freedom, is the complicated structure of energy spectrum.\nWe shall consider quantum mechanical-system with parametrically dependent\nenergy spectrum. In particular, we study energy spectrum of the\nMathieu-Schrodinger equation. Structure of the spectrum is quite non-trivial,\nconsists from the domains of non-degenerated and degenerated stats, separated\nfrom each other by branch points.", + "category": "nlin_CD" + }, + { + "text": "Fast Ensemble Smoothing: Smoothing is essential to many oceanographic, meteorological and hydrological\napplications. The interval smoothing problem updates all desired states within\na time interval using all available observations. The fixed-lag smoothing\nproblem updates only a fixed number of states prior to the observation at\ncurrent time. The fixed-lag smoothing problem is, in general, thought to be\ncomputationally faster than a fixed-interval smoother, and can be an\nappropriate approximation for long interval-smoothing problems. In this paper,\nwe use an ensemble-based approach to fixed-interval and fixed-lag smoothing,\nand synthesize two algorithms. The first algorithm produces a linear time\nsolution to the interval smoothing problem with a fixed factor, and the second\none produces a fixed-lag solution that is independent of the lag length.\nIdentical-twin experiments conducted with the Lorenz-95 model show that for lag\nlengths approximately equal to the error doubling time, or for long intervals\nthe proposed methods can provide significant computational savings. These\nresults suggest that ensemble methods yield both fixed-interval and fixed-lag\nsmoothing solutions that cost little additional effort over filtering and model\npropagation, in the sense that in practical ensemble application the additional\nincrement is a small fraction of either filtering or model propagation costs.\nWe also show that fixed-interval smoothing can perform as fast as fixed-lag\nsmoothing and may be advantageous when memory is not an issue.", + "category": "nlin_CD" + }, + { + "text": "Colloquium: Theory of Drag Reduction by Polymers in Wall Bounded\n Turbulence: The flow of fluids in channels, pipes or ducts, as in any other wall-bounded\nflow (like water along the hulls of ships or air on airplanes) is hindered by a\ndrag, which increases many-folds when the fluid flow turns from laminar to\nturbulent. A major technological problem is how to reduce this drag in order to\nminimize the expense of transporting fluids like oil in pipelines, or to move\nships in the ocean. It was discovered in the mid-twentieth century that minute\nconcentrations of polymers can reduce the drag in turbulent flows by up to 80%.\nWhile experimental knowledge had accumulated over the years, the fundamental\ntheory of drag reduction by polymers remained elusive for a long time, with\narguments raging whether this is a \"skin\" or a \"bulk\" effect. In this\ncolloquium review we first summarize the phenomenology of drag reduction by\npolymers, stressing both its universal and non-universal aspects, and then\nproceed to review a recent theory that provides a quantitative explanation of\nall the known phenomenology. We treat both flexible and rod-like polymers,\nexplaining the existence of universal properties like the Maximum Drag\nReduction (MDR) asymptote, as well as non-universal cross-over phenomena that\ndepend on the Reynolds number, on the nature of the polymer and on its\nconcentration. Finally we also discuss other agents for drag reduction with a\nstress on the important example of bubbles.", + "category": "nlin_CD" + }, + { + "text": "Acceleration and vortex filaments in turbulence: We report recent results from a high resolution numerical study of fluid\nparticles transported by a fully developed turbulent flow. Single particle\ntrajectories were followed for a time range spanning more than three decades,\nfrom less than a tenth of the Kolmogorov time-scale up to one large-eddy\nturnover time. We present some results concerning acceleration statistics and\nthe statistics of trapping by vortex filaments.", + "category": "nlin_CD" + }, + { + "text": "Response and Sensitivity Using Markov Chains: Dynamical systems are often subject to forcing or changes in their governing\nparameters and it is of interest to study how this affects their statistical\nproperties. A prominent real-life example of this class of problems is the\ninvestigation of climate response to perturbations. In this respect, it is\ncrucial to determine what the linear response of a system is to small\nperturbations as a quantification of sensitivity. Alongside previous work, here\nwe use the transfer operator formalism to study the response and sensitivity of\na dynamical system undergoing perturbations. By projecting the transfer\noperator onto a suitable finite dimensional vector space, one is able to obtain\nmatrix representations which determine finite Markov processes. Further, using\nperturbation theory for Markov matrices, it is possible to determine the linear\nand nonlinear response of the system given a prescribed forcing. Here, we\nsuggest a methodology which puts the scope on the evolution law of densities\n(the Liouville/Fokker-Planck equation), allowing to effectively calculate the\nsensitivity and response of two representative dynamical systems.", + "category": "nlin_CD" + }, + { + "text": "Phase space structure and escape time dynamics in a Van der Waals model\n for exothermic reactions: We study the phase space structures that control the transport in a classical\nHamiltonian model for a chemical reaction. This model has been proposed to\nstudy the yield of products in an ultracold exothermic reaction. In the\nconsidered model, two elements determine the evolution of the system: a Van der\nWaals force and short-range force associated with the many-body interactions.\nIn the previous work has been used small random periodic changes in the\ndirection of the momentum to simulate the short-range many-body interactions.\nIn the present work, random Gaussian bumps have been added to the Van der Waals\npotential energy simulate the short-range effects between the particles in the\nsystem. We compare both variants of the model and explain their differences\nsimilarities and differences from a phase space perspective. In order to\nvisualize the structures that direct the dynamics in the phase space, we\nconstruct a natural Lagrangian descriptor for Hamiltonian systems based on the\nMaupertuis action $S_0$.", + "category": "nlin_CD" + }, + { + "text": "Breaking time reversal symmetry in chaotic driven Rydberg atoms: We consider the dynamics of Rydberg states of the hydrogen atom driven by a\nmicrowave field of elliptical polarization, with a possible additional static\nelectric field. We concentrate on the effect of a resonant weak field - whose\nfrequency is close to the Kepler frequency of the electron around the nucleus -\nwhich essentially produces no ionization of the atom, but completely mixes the\nvarious states inside an hydrogenic manifold of fixed principal quantum number.\nFor sufficiently small fields, a perturbative approach (both in classical and\nquantum mechanics) is relevant. For some configurations of the fields, the\nclassical secular motion (i.e. evolution in time of the elliptical electronic\ntrajectory) is shown to be predominantly chaotic. Changing the orientation of\nthe static field with respect to the polarization of the microwave field allows\nus to investigate the effect of generalized time-reversal symmetry breaking on\nthe statistical properties of energy levels.", + "category": "nlin_CD" + }, + { + "text": "Quantum signatures of classical multifractal measures: A clear signature of classical chaoticity in the quantum regime is the\nfractal Weyl law, which connects the density of eigenstates to the dimension\n$D_0$ of the classical invariant set of open systems. Quantum systems of\ninterest are often {\\it partially} open (e.g., cavities in which trajectories\nare partially reflected/absorbed). In the corresponding classical systems $D_0$\nis trivial (equal to the phase-space dimension), and the fractality is\nmanifested in the (multifractal) spectrum of R\\'enyi dimensions $D_q$. In this\npaper we investigate the effect of such multifractality on the Weyl law. Our\nnumerical simulations in area-preserving maps show for a wide range of\nconfigurations and system sizes $M$ that (i) the Weyl law is governed by a\ndimension different from $D_0=2$ and (ii) the observed dimension oscillates as\na function of $M$ and other relevant parameters. We propose a classical model\nwhich considers an undersampled measure of the chaotic invariant set, explains\nour two observations, and predicts that the Weyl law is governed by a\nnon-trivial dimension $D_\\mathrm{asymptotic} < D_0$ in the semi-classical limit\n$M\\rightarrow\\infty$.", + "category": "nlin_CD" + }, + { + "text": "Non-Gaussian Statistics of Multiple Filamentation: We consider the statistics of light amplitude fluctuations for the\npropagation of a laser beam subjected to multiple filamentation in an amplified\nKerr media, with both linear and nonlinear dissipation. Dissipation arrests the\ncatastrophic collapse of filaments, causing their disintegration into almost\nlinear waves. These waves form a nearly-Gaussian random field which seeds new\nfilaments. For small amplitudes the probability density function (PDF) of light\namplitude is close to Gaussian, while for large amplitudes the PDF has a long\npower-like tail which corresponds to strong non-Gaussian fluctuations, i.e.\nintermittency of strong optical turbulence. This tail is determined by the\nuniversal form of near singular filaments and the PDF for the maximum\namplitudes of the filaments.", + "category": "nlin_CD" + }, + { + "text": "Chaotic behavior of ion exchange phenomena in polymer gel electrolytes\n through irradiated polymeric membrane: A desktop experiment has been done to show the nonlinearity in the I-V\ncharacteristics of an ion conducting electrochemical micro-system. Its chaotic\ndynamics is being reported for the first time which has been captured by an\nelectronic circuit. Polyvinylidene fluoride-co-hexafluoropropene (PVdF-HFP) gel\nelectrolyte comprising of a combination of plasticizers (ethylene carbonate and\npropylene carbonate) and salts have been prepared to study the exchange of ions\nthrough porous poly ethylene terephthalate (PET) membranes. The nonlinearity of\nthis system is due to the ion exchange of the polymer gel electrolytes (PGEs)\nthrough a porous membrane. The different regimes of spiking and non-spiking\nchaotic motions are being presented. The possible applications are highlighted.", + "category": "nlin_CD" + }, + { + "text": "Asymmetry of temporal cross-correlations in turbulent shear flows: We investigate spatial and temporal cross-correlations between streamwise and\nnormal velocity components in three shear flows: a low-dimensional model for\nvortex-streak interactions, direct numerical simulations for a nearly\nhomogeneous shear flow and experimental data for a turbulent boundary layer. A\ndriving of streamwise streaks by streamwise vortices gives rise to a temporal\nasymmetry in the short time correlation. Close to the wall or the bounding\nsurface in the free-slip situations, this asymmetry is identified. Further away\nfrom the boundaries the asymmetry becomes weaker and changes character,\nindicating the prevalence of other processes. The systematic variation of the\nasymmetry measure may be used as a complementary indicator to separate\ndifferent layers in turbulent shear flows. The location of the extrema at\ndifferent streamwise displacements can be used to read off the mean advection\nspeed; it differs from the mean streamwise velocity because of asymmetries in\nthe normal extension of the structures.", + "category": "nlin_CD" + }, + { + "text": "Optimal conditions for the numerical calculation of the largest Lyapunov\n exponent for systems of ordinary differential equations: A general indicator of the presence of chaos in a dynamical system is the\nlargest Lyapunov exponent. This quantity provides a measure of the mean\nexponential rate of divergence of nearby orbits. In this paper, we show that\nthe so-called two-particle method introduced by Benettin et al. could lead to\nspurious estimations of the largest Lyapunov exponent. As a comparator method,\nthe maximum Lyapunov exponent is computed from the solution of the variational\nequations of the system. We show that the incorrect estimation of the largest\nLyapunov exponent is based on the setting of the renormalization time and the\ninitial distance between trajectories. Unlike previously published works, we\nhere present three criteria that could help to determine correctly these\nparameters so that the maximum Lyapunov exponent is close to the expected\nvalue. The results have been tested with four well known dynamical systems:\nUeda, Duffing, R\\\"ossler and Lorenz.", + "category": "nlin_CD" + }, + { + "text": "Energy Transfer and Coherence in Coupled Oscillators with Delayed\n Coupling: A Classical Picture for Two-Level Systems: The Frimmer-Novotny model to simulate two-level systems by coupled\noscillators is extended by incorporating a constant time delay in the coupling.\nThe effects of the introduced delay on system dynamics and two-level modeling\nare then investigated and found substantial. Mathematically, introducing a\ndelay converts the dynamical system from a finite one into an\ninfinite-dimensional system. The resulted system of delay differential\nequations is solved using the Krylov method with Chebyshev interpolation and\npost-processing refinement. The calculations and analyses reveal the critical\nrole that a delay can play. It has oscillatory effects as the main dynamical\neigenmodes move around a circle with a radius proportional to the coupling\nstrength and an angle linear with the delay. This alteration governs the energy\ntransfer dynamics and coherence. Accordingly, both, the delay and the coupling\nstrength dictate the stability of the system. The delay is the main related\nparameter as for certain intervals of it, the system remains stable regardless\nof the coupling. A significant effect occurs when one of the main modes crosses\nthe imaginary axis, where it becomes pure imaginary and dampingless. Thus, the\ntwo states energies can live and be exchanged for an extremely long time.\nFurthermore, it is found that the delay alters both the splitting and the\nlinewidth in a way further influencing the energy transfer and coherence. It is\nfound also that the delay should not be large to have significant effect. For\nexample, for an optical system with 500 nm wavelength, the critical delay can\nbe in tens of attoseconds.", + "category": "nlin_CD" + }, + { + "text": "Exact Synchronization for Finite-State Sources: We analyze how an observer synchronizes to the internal state of a\nfinite-state information source, using the epsilon-machine causal\nrepresentation. Here, we treat the case of exact synchronization, when it is\npossible for the observer to synchronize completely after a finite number of\nobservations. The more difficult case of strictly asymptotic synchronization is\ntreated in a sequel. In both cases, we find that an observer, on average, will\nsynchronize to the source state exponentially fast and that, as a result, the\naverage accuracy in an observer's predictions of the source output approaches\nits optimal level exponentially fast as well. Additionally, we show here how to\nanalytically calculate the synchronization rate for exact epsilon-machines and\nprovide an efficient polynomial-time algorithm to test epsilon-machines for\nexactness.", + "category": "nlin_CD" + }, + { + "text": "About ergodicity in the family of limacon billiards: By continuation from the hyperbolic limit of the cardioid billiard we show\nthat there is an abundance of bifurcations in the family of limacon billiards.\nThe statistics of these bifurcation shows that the size of the stable intervals\ndecreases with approximately the same rate as their number increases with the\nperiod. In particular, we give numerical evidence that arbitrarily close to the\ncardioid there are elliptic islands due to orbits created in saddle node\nbifurcations. This shows explicitly that if in this one parameter family of\nmaps ergodicity occurs for more than one parameter the set of these parameter\nvalues has a complicated structure.", + "category": "nlin_CD" + }, + { + "text": "Regularized Renormalization Group Reduction of Symplectic Map: By means of the perturbative renormalization group method, we study a\nlong-time behaviour of some symplectic discrete maps near elliptic and\nhyperbolic fixed points. It is shown that a naive renormalization group (RG)\nmap breaks the symplectic symmetry and fails to describe a long-time behaviour.\nIn order to preserve the symplectic symmetry, we present a regularization\nprocedure, which gives a regularized symplectic RG map describing an\napproximate long-time behaviour succesfully.", + "category": "nlin_CD" + }, + { + "text": "Fractional Maps and Fractional Attractors. Part II: Fractional\n Difference $\u03b1$-Families of Maps: In this paper we extend the notion of an $\\alpha$-family of maps to discrete\nsystems defined by simple difference equations with the fractional Caputo\ndifference operator. The equations considered are equivalent to maps with\nfalling factorial-law memory which is asymptotically power-law memory. We\nintroduce the fractional difference Universal, Standard, and Logistic\n$\\alpha$-Families of Maps and propose to use them to study general properties\nof discrete nonlinear systems with asymptotically power-law memory.", + "category": "nlin_CD" + }, + { + "text": "Exact Spectral Form Factor in a Minimal Model of Many-Body Quantum Chaos: The most general and versatile defining feature of quantum chaotic systems is\nthat they possess an energy spectrum with correlations universally described by\nrandom matrix theory (RMT). This feature can be exhibited by systems with a\nwell defined classical limit as well as by systems with no classical\ncorrespondence, such as locally interacting spins or fermions. Despite great\nphenomenological success, a general mechanism explaining the emergence of RMT\nwithout reference to semiclassical concepts is still missing. Here we provide\nthe example of a quantum many-body system with no semiclassical limit (no large\nparameter) where the emergence of RMT spectral correlations is proven exactly.\nSpecifically, we consider a periodically driven Ising model and write the\nFourier transform of spectral density's two-point function, the spectral form\nfactor, in terms of a partition function of a two-dimensional classical Ising\nmodel featuring a space-time duality. We show that the self-dual cases provide\na minimal model of many-body quantum chaos, where the spectral form factor is\ndemonstrated to match RMT for all values of the integer time variable $t$ in\nthe thermodynamic limit. In particular, we rigorously prove RMT form factor for\nodd $t$, while we formulate a precise conjecture for even $t$. The results\nimply ergodicity for any finite amount of disorder in the longitudinal field,\nrigorously excluding the possibility of many-body localization. Our method\nprovides a novel route for obtaining exact nonperturbative results in\nnon-integrable systems.", + "category": "nlin_CD" + }, + { + "text": "On the Newton-Raphson basins of convergence associated with the\n libration points in the axisymmetric five-body problem: the concave\n configuration: The axisymmetric five-body problem with the concave configuration has been\nstudied numerically to reveal the basins of convergence, by exploring the\nNewton-Raphson iterative scheme, corresponding to the coplanar libration points\n(which act as attractors). In addition, four primaries are set in axisymmetric\ncentral configurations introduced by \\'{E}rdi and Czirj\\'{a}k and the motion is\ngoverned by mutual gravitational attraction only. The evolution of the\npositions of libration points is illustrated, as a function of the value of\nangle parameters. A systematic and rigorous investigation is performed in an\neffort to unveil how the angle parameters affect the topology of the basins of\nconvergence. In addition, the relation of the domain of basins of convergence\nwith required number of iterations and the corresponding probability\ndistributions are illustrated.", + "category": "nlin_CD" + }, + { + "text": "Multifaceted nonlinear dynamics in $\\mathcal{PT}$-symmetric coupled\n Li\u00e9nard oscillators: We propose a generalized parity-time ($\\mathcal{PT}$) -symmetric Li\\'enard\noscillator with two different orders of nonlinear position-dependent\ndissipation. We study the stability of the stationary states by using the\neigenvalues of Jacobian and evaluate the stability threshold thereafter. In the\nfirst order nonlinear damping model, we discover that the temporal evolution of\nboth gain and lossy oscillators attains a complete convergence towards the\nstable stationary state leading to the emergence of oscillation and amplitude\ndeaths. Also, the system displays a remarkable manifestation of transient chaos\nin the lossy oscillator while the gain counterpart exhibits blow-up dynamics\nfor certain choice of initial conditions and control parameters. Employing an\nexternal driving force on the loss oscillator, we find that the blow-up\ndynamics can be controlled and a pure aperiodic state is achievable. On the\nother hand, the second order nonlinear damping model yields a completely\ndifferent dynamics on contrary to the first order where the former reveals a\nconventional quasi-periodic route to chaos upon decreasing the natural\nfrequency of both gain and loss oscillators. An electronic circuit scheme for\nthe experimental realization of the proposed system has also been put forward.", + "category": "nlin_CD" + }, + { + "text": "Influence of symmetry breaking on the fluctuation properties of spectra: We study the effect of gradual symmetry breaking in a non-integrable system\non the level fluctuation statistics. We consider the case when the symmetry is\nrepresented by a quantum number that takes one of two possible values, so that\nthe unperturbed system has a spectrum composed of two independent sequences.\nWhen symmetry-breaking perturbation is represented by a random matrix with an\nadjustable strength, the shape of the spectrum monotonously evolves towards the\nWigner distribution as the strength parameter increases. This contradicts the\nobserved behaviour of the acoustic resonance spectra in quartz blocks during\nthe breaking of a point-group symmetry that has two eigenvalues, where the\nsystem changes in the beginning towards the Poisson statistics then turns back\nto the GOE statistics. This behaviour is explained by assuming that the\nsymmetry breaking perturbation removes the degeneracy of a limited number of\nlevels, thus creating a third chaotic sequence. As symmetry breaking increases,\nthe new sequence grows at the expense of the initial pair until it overwhelms\nthe whole spectrum when the symmetry completely disappears. The calculated\nspacing distribution and spectral rigidity are able to describe the evolution\nof the observed acoustic resonance spectra.", + "category": "nlin_CD" + }, + { + "text": "Hidden attractors in fundamental problems and engineering models: Recently a concept of self-excited and hidden attractors was suggested: an\nattractor is called a self-excited attractor if its basin of attraction\noverlaps with neighborhood of an equilibrium, otherwise it is called a hidden\nattractor. For example, hidden attractors are attractors in systems with no\nequilibria or with only one stable equilibrium (a special case of\nmultistability and coexistence of attractors). While coexisting self-excited\nattractors can be found using the standard computational procedure, there is no\nstandard way of predicting the existence or coexistence of hidden attractors in\na system. In this plenary survey lecture the concept of self-excited and hidden\nattractors is discussed, and various corresponding examples of self-excited and\nhidden attractors are considered.", + "category": "nlin_CD" + }, + { + "text": "Flow induced ultrasound scattering: experimental studies: Sound scattering by a finite width beam on a single rigid body rotation\nvortex flow is detected by a linear array of transducers (both smaller than a\nflow cell), and analyzed using a revised scattering theory. Both the phase and\namplitude of the scattered signal are obtained on 64 elements of the detector\narray and used for the analysis of velocity and vorticity fields. Due to\naveraging on many pulses the signal-to-noise ratio of the phases difference in\nthe scattered sound signal can be amplified drastically, and the resolution of\nthe method in the detection of circulation, vortex radius, vorticity, and\nvortex location becomes comparable with that obtained earlier by time-reversal\nmirror (TRM) method (P. Roux, J. de Rosny, M. Tanter, and M. Fink, {\\sl Phys.\nRev. Lett.} {\\bf 79}, 3170 (1997)). The revised scattering theory includes two\ncrucial steps, which allow overcoming limitations of the existing theories.\nFirst, the Huygens construction of a far field scattering signal is carried out\nfrom a signal obtained at any intermediate plane. Second, a beam function that\ndescribes a finite width beam is introduced, which allows using a theory\ndeveloped for an infinite width beam for the relation between a scattering\namplitude and the vorticity structure function. Structure functions of the\nvelocity and vorticity fields deduced from the sound scattering signal are\ncompared with those obtained from simultaneous particle image velocimetry (PIV)\nmeasurements. Good quantitative agreement is found.", + "category": "nlin_CD" + }, + { + "text": "Universality of attractors at weak dissipation and particles\n distribution in turbulence: We study stationary solutions to the continuity equation for weakly\ncompressible flows. These describe non-equilibrium steady states of weakly\ndissipative dynamical systems. Compressibility is a singular perturbation that\nchanges the steady state density from a constant \"microcanonical\" distribution\ninto a singular multifractal measure supported on the \"strange attractor\". We\nintroduce a representation of the latter and show that the space-averaged\nproperties are described universally by a log-normal distribution determined by\na single structure function. The spectrum of fractal dimensions is derived.\nApplication to the problem of distribution of particles in turbulence gives\ntestable predictions for real turbulence and stresses the role of pressure\nfluctuations.", + "category": "nlin_CD" + }, + { + "text": "Oscillations and synchronization in a system of three reactively coupled\n oscillators: We consider a system of three interacting van der Pol oscillators with\nreactive coupling. Phase equations are derived, using proper order of expansion\nover the coupling parameter. The dynamics of the system is studied by means of\nthe bifurcation analysis and with the method of Lyapunov exponent charts.\nEssential and physically meaningful features of the reactive coupling are\ndiscussed.", + "category": "nlin_CD" + }, + { + "text": "A Topological Study of Chaotic Iterations. Application to Hash Functions: Chaotic iterations, a tool formerly used in distributed computing, has\nrecently revealed various interesting properties of disorder leading to its use\nin the computer science security field. In this paper, a comprehensive study of\nits topological behavior is proposed. It is stated that, in addition to being\nchaotic as defined in the Devaney's formulation, this tool possesses the\nproperty of topological mixing. Additionally, its level of sensibility,\nexpansivity, and topological entropy are evaluated. All of these properties\nlead to a complete unpredictable behavior for the chaotic iterations. As it\nonly manipulates binary digits or integers, we show that it is possible to use\nit to produce truly chaotic computer programs. As an application example, a\ntruly chaotic hash function is proposed in two versions. In the second version,\nan artificial neural network is used, which can be stated as chaotic according\nto Devaney.", + "category": "nlin_CD" + }, + { + "text": "Chaotic synchronization induced by external noise in coupled limit cycle\n oscillators: A solvable model of noise effects on globally coupled limit cycle oscillators\nis proposed. The oscillators are under the influence of independent and\nadditive white Gaussian noise. The averaged motion equation of the system with\ninfinitely coupled oscillators is derived without any approximation through an\nanalysis based on the nonlinear Fokker--Planck equation. Chaotic\nsynchronization associated with the appearance of macroscopic chaotic behavior\nis shown by investigating the changes in averaged motion with increasing noise\nintensity.", + "category": "nlin_CD" + }, + { + "text": "Understanding complex dynamics by means of an associated Riemann surface: We provide an example of how the complex dynamics of a recently introduced\nmodel can be understood via a detailed analysis of its associated Riemann\nsurface. Thanks to this geometric description an explicit formula for the\nperiod of the orbits can be derived, which is shown to depend on the initial\ndata and the continued fraction expansion of a simple ratio of the coupling\nconstants of the problem. For rational values of this ratio and generic values\nof the initial data, all orbits are periodic and the system is isochronous. For\nirrational values of the ratio, there exist periodic and quasi-periodic orbits\nfor different initial data. Moreover, the dependence of the period on the\ninitial data shows a rich behavior and initial data can always be found such\nthe period is arbitrarily high.", + "category": "nlin_CD" + }, + { + "text": "Consequences of Flooding on Spectral Statistics: We study spectral statistics in systems with a mixed phase space, in which\nregions of regular and chaotic motion coexist. Increasing their density of\nstates, we observe a transition of the level-spacing distribution P(s) from\nBerry-Robnik to Wigner statistics, although the underlying classical\nphase-space structure and the effective Planck constant remain unchanged. This\ntransition is induced by flooding, i.e., the disappearance of regular states\ndue to increasing regular-to-chaotic couplings. We account for this effect by a\nflooding-improved Berry-Robnik distribution, in which an effectively reduced\nsize of the regular island enters. To additionally describe power-law level\nrepulsion at small spacings, we extend this prediction by explicitly\nconsidering the tunneling couplings between regular and chaotic states. This\nresults in a flooding- and tunneling-improved Berry-Robnik distribution which\nis in excellent agreement with numerical data.", + "category": "nlin_CD" + }, + { + "text": "On possibility of realization of the phenomena of complex analytic\n dynamics in physical systems. Novel mechanism of the synchronization loss in\n coupled period-doubling systems: The possibility of realization of the phenomena of complex analytic dynamics\nfor the realistic physical models are investigated. Observation of the\nMandelbrot and Julia sets in the parameter and phase spaces both for the\ndiscrete maps and non-autonomous continuous systems is carried out. For these\npurposes, the method, based on consideration of coupled systems, demonstrating\nperiod-doubling cascade is suggested. Novel mechanism of synchronization loss\nin coupled systems corresponded to the dynamical behavior intrinsic to the\ncomplex analytic maps is offered.", + "category": "nlin_CD" + }, + { + "text": "Testing Dynamical System Variables for Reconstruction: Analyzing data from dynamical systems often begins with creating a\nreconstruction of the trajectory based on one or more variables, but not all\nvariables are suitable for reconstructing the trajectory. The concept of\nnonlinear observability has been investigated as a way to determine if a\ndynamical system can be reconstructed from one signal or a combination of\nsignals, however nonlinear observability can be difficult to calculate for a\nhigh dimensional system. In this work I compare the results from nonlinear\nobservability to a continuity statistic that indicates the likelihood that\nthere is a continuous function between two sets of multidimensional points- in\nthis case two different reconstructions of the same attractor from different\nsignals simultaneously measured.\n Without a metric against which to test the ability to reconstruct a system,\nthe predictions of nonlinear observability and continuity are ambiguous. As a\nadditional test how well different signals can predict the ability to\nreconstruct a dynamical system I use the fitting error from training a\nreservoir computer.", + "category": "nlin_CD" + }, + { + "text": "Inferring unobserved multistrain epidemic sub-populations using\n synchronization dynamics: A new method is proposed to infer unobserved epidemic sub-populations by\nexploiting the synchronization properties of multistrain epidemic models. A\nmodel for dengue fever is driven by simulated data from secondary infective\npopulations. Primary infective populations in the driven system synchronize to\nthe correct values from the driver system. Most hospital cases of dengue are\nsecondary infections, so this method provides a way to deduce unobserved\nprimary infection levels. We derive center manifold equations that relate the\ndriven system to the driver system and thus motivate the use of synchronization\nto predict unobserved primary infectives. Synchronization stability between\nprimary and secondary infections is demonstrated through numerical measurements\nof conditional Lyapunov exponents and through time series simulations.", + "category": "nlin_CD" + }, + { + "text": "Weak multiplexing in neural networks: Switching between chimera and\n solitary states: We investigate spatio-temporal patterns occurring in a two-layer multiplex\nnetwork of oscillatory FitzHugh-Nagumo neurons, where each layer is represented\nby a nonlocally coupled ring. We show that weak multiplexing, i.e., when the\ncoupling between the layers is smaller than that within the layers, can have a\nsignificant impact on the dynamics of the neural network. We develop control\nstrategies based on weak multiplexing and demonstrate how the desired state in\none layer can be achieved without manipulating its parameters, but only by\nadjusting the other layer. We find that for coupling range mismatch weak\nmultiplexing leads to the appearance of chimera states with different shapes of\nthe mean velocity profile for parameter ranges where they do not exist in\nisolation. Moreover, we show that introducing a coupling strength mismatch\nbetween the layers can suppress chimera states with one incoherent domain\n(one-headed chimeras) and induce various other regimes such as in-phase\nsynchronization or two-headed chimeras. Interestingly, small intra-layer\ncoupling strength mismatch allows to achieve solitary states throughout the\nwhole network.", + "category": "nlin_CD" + }, + { + "text": "Non-hyperbolicity at large scales of a high-dimensional chaotic system: The dynamics of many important high-dimensional dynamical systems are both\nchaotic and complex, meaning that strong reducing hypotheses are required to\nunderstand the dynamics. The highly influential chaotic hypothesis of\nGallavotti and Cohen states that the large-scale dynamics of high-dimensional\nsystems are effectively uniformly hyperbolic, which implies many felicitous\nstatistical properties. We obtain direct and reliable numerical evidence,\ncontrary to the chaotic hypothesis, of the existence of non-hyperbolic\nlarge-scale dynamical structures in a mean-field coupled system. To do this we\nreduce the system to its thermodynamic limit, which we approximate numerically\nwith a Chebyshev basis transfer operator discretisation. This enables us to\nobtain a high precision estimate of a homoclinic tangency, implying a failure\nof uniform hyperbolicity. Robust non-hyperbolic behaviour is expected under\nperturbation. As a result, the chaotic hypothesis should not be {\\it a priori}\nassumed to hold in all systems, and a better understanding of the domain of its\nvalidity is required.", + "category": "nlin_CD" + }, + { + "text": "Local softening of information geometric indicators of chaos in\n statistical modeling in the presence of quantum-like considerations: In a previous paper (C. Cafaro et al., 2012), we compared an uncorrelated 3D\nGaussian statistical model to an uncorrelated 2D Gaussian statistical model\nobtained from the former model by introducing a constraint that resembles the\nquantum mechanical canonical minimum uncertainty relation. Analysis was\ncompleted by way of the information geometry and the entropic dynamics of each\nsystem. This analysis revealed that the chaoticity of the 2D Gaussian\nstatistical model, quantified by means of the Information Geometric Entropy\n(IGE), is softened or weakened with respect to the chaoticity of the 3D\nGaussian statistical model due to the accessibility of more information. In\nthis companion work, we further constrain the system in the context of a\ncorrelation constraint among the system's micro-variables and show that the\nchaoticity is further weakened, but only locally. Finally, the physicality of\nthe constraints is briefly discussed, particularly in the context of quantum\nentanglement.", + "category": "nlin_CD" + }, + { + "text": "Jets, Stickiness and Anomalous Transport: Dynamical and statistical properties of the vortex and passive particle\nadvection in chaotic flows generated by four and sixteen point vortices are\ninvestigated. General transport properties of these flows are found anomalous\nand exhibit a superdiffusive behavior with typical second moment exponent (\\mu\n\\sim 1.75). The origin of this anomaly is traced back to the presence of\ncoherent structures within the flow, the vortex cores and the region far from\nwhere vortices are located. In the vicinity of these regions stickiness is\nobserved and the motion of tracers is quasi-ballistic. The chaotic nature of\nthe underlying flow dictates the choice for thorough analysis of transport\nproperties. Passive tracer motion is analyzed by measuring the mutual relative\nevolution of two nearby tracers. Some tracers travel in each other vicinity for\nrelatively large times. This is related to an hidden order for the tracers\nwhich we call jets. Jets are localized and found in sticky regions. Their\nstructure is analyzed and found to be formed of a nested sets of jets within\njets. The analysis of the jet trapping time statistics shows a quantitative\nagreement with the observed transport exponent.", + "category": "nlin_CD" + }, + { + "text": "Manifold Learning Approach for Chaos in the Dripping Faucet: Dripping water from a faucet is a typical example exhibiting rich nonlinear\nphenomena. For such a system, the time stamps at which water drops separate\nfrom the faucet can be directly observed in real experiments, and the time\nseries of intervals \\tau_n between drop separations becomes a subject of\nanalysis. Even if the mass m_n of a drop at the onset of the n-th separation,\nwhich cannot be observed directly, exhibits perfectly deterministic dynamics,\nit sometimes fails to obtain important information from time series of \\tau_n.\nThis is because the return plot \\tau_n-1 vs. \\tau_n may become a multi-valued\nfunction, i.e., not a deterministic dynamical system. In this paper, we propose\na method to construct a nonlinear coordinate which provides a \"surrogate\" of\nthe internal state m_n from the time series of \\tau_n. Here, a key of the\nproposed approach is to use ISOMAP, which is a well-known method of manifold\nlearning. We first apply it to the time series of $\\tau_n$ generated from the\nnumerical simulation of a phenomenological mass-spring model for the dripping\nfaucet system. It is shown that a clear one-dimensional map is obtained by the\nproposed approach, whose characteristic quantities such as the Lyapunov\nexponent, the topological entropy, and the time correlation function coincide\nwith the original dripping faucet system. Furthermore, we also analyze data\nobtained from real dripping faucet experiments which also provides promising\nresults.", + "category": "nlin_CD" + }, + { + "text": "Nonlinear Hamiltonian dynamics of Lagrangian transport and mixing in the\n ocean: Methods of dynamical system's theory are used for numerical study of\ntransport and mixing of passive particles (water masses, temperature, salinity,\npollutants, etc.) in simple kinematic ocean models composed with the main\nEulerian coherent structures in a randomly fluctuating ocean -- a jet-like\ncurrent and an eddy. Advection of passive tracers in a periodically-driven flow\nconsisting of a background stream and an eddy (the model inspired by the\nphenomenon of topographic eddies over mountains in the ocean and atmosphere) is\nanalyzed as an example of chaotic particle's scattering and transport. A\nnumerical analysis reveals a nonattracting chaotic invariant set $\\Lambda$ that\ndetermines scattering and trapping of particles from the incoming flow. It is\nshown that both the trapping time for particles in the mixing region and the\nnumber of times their trajectories wind around the vortex have hierarchical\nfractal structure as functions of the initial particle's coordinates.\nScattering functions are singular on a Cantor set of initial conditions, and\nthis property should manifest itself by strong fluctuations of quantities\nmeasured in experiments. The Lagrangian structures in our numerical experiments\nare shown to be similar to those found in a recent laboratory dye experiment at\nWoods Hole. Transport and mixing of passive particles is studied in the\nkinematic model inspired by the interaction of a jet current (like the Gulf\nStream or the Kuroshio) with an eddy in a noisy environment. We demonstrate a\nnon-trivial phenomenon of noise-induced clustering of passive particles and\npropose a method to find such clusters in numerical experiments. These clusters\nare patches of advected particles which can move together in a random velocity\nfield for comparatively long time.", + "category": "nlin_CD" + }, + { + "text": "Eigenfunction entropy and spectral compressibility for critical random\n matrix ensembles: Based on numerical and perturbation series arguments we conjecture that for\ncertain critical random matrix models the information dimension of\neigenfunctions D_1 and the spectral compressibility chi are related by the\nsimple equation chi+D_1/d=1, where d is the system dimensionality.", + "category": "nlin_CD" + }, + { + "text": "Strong and weak chaos in weakly nonintegrable many-body Hamiltonian\n systems: We study properties of chaos in generic one-dimensional nonlinear Hamiltonian\nlattices comprised of weakly coupled nonlinear oscillators, by numerical\nsimulations of continuous-time systems and symplectic maps. For small coupling,\nthe measure of chaos is found to be proportional to the coupling strength and\nlattice length, with the typical maximal Lyapunov exponent being proportional\nto the square root of coupling. This strong chaos appears as a result of\ntriplet resonances between nearby modes. In addition to strong chaos we observe\na weakly chaotic component having much smaller Lyapunov exponent, the measure\nof which drops approximately as a square of the coupling strength down to\nsmallest couplings we were able to reach. We argue that this weak chaos is\nlinked to the regime of fast Arnold diffusion discussed by Chirikov and\nVecheslavov. In disordered lattices of large size we find a subdiffusive\nspreading of initially localized wave packets over larger and larger number of\nmodes. The relations between the exponent of this spreading and the exponent in\nthe dependence of the fast Arnold diffusion on coupling strength are analyzed.\nWe also trace parallels between the slow spreading of chaos and deterministic\nrheology.", + "category": "nlin_CD" + }, + { + "text": "Lyapunov spectra of chaotic recurrent neural networks: Brains process information through the collective dynamics of large neural\nnetworks. Collective chaos was suggested to underlie the complex ongoing\ndynamics observed in cerebral cortical circuits and determine the impact and\nprocessing of incoming information streams. In dissipative systems, chaotic\ndynamics takes place on a subset of phase space of reduced dimensionality and\nis organized by a complex tangle of stable, neutral and unstable manifolds. Key\ntopological invariants of this phase space structure such as attractor\ndimension, and Kolmogorov-Sinai entropy so far remained elusive.\n Here we calculate the complete Lyapunov spectrum of recurrent neural\nnetworks. We show that chaos in these networks is extensive with a\nsize-invariant Lyapunov spectrum and characterized by attractor dimensions much\nsmaller than the number of phase space dimensions. We find that near the onset\nof chaos, for very intense chaos, and discrete-time dynamics, random matrix\ntheory provides analytical approximations to the full Lyapunov spectrum. We\nshow that a generalized time-reversal symmetry of the network dynamics induces\na point-symmetry of the Lyapunov spectrum reminiscent of the symplectic\nstructure of chaotic Hamiltonian systems. Fluctuating input reduces both the\nentropy rate and the attractor dimension. For trained recurrent networks, we\nfind that Lyapunov spectrum analysis provides a quantification of error\npropagation and stability achieved. Our methods apply to systems of arbitrary\nconnectivity, and we describe a comprehensive set of controls for the accuracy\nand convergence of Lyapunov exponents.\n Our results open a novel avenue for characterizing the complex dynamics of\nrecurrent neural networks and the geometry of the corresponding chaotic\nattractors. They also highlight the potential of Lyapunov spectrum analysis as\na diagnostic for machine learning applications of recurrent networks.", + "category": "nlin_CD" + }, + { + "text": "Chaos and localization in the Discrete Nonlinear Schr\u00f6dinger Equation: We analyze the chaotic dynamics of a one-dimensional discrete nonlinear\nSchr\\\"odinger equation. This nonintegrable model, ubiquitous in several fields\nof physics, describes the behavior of an array of coupled complex oscillators\nwith a local nonlinear potential. We explore the Lyapunov spectrum for\ndifferent values of the energy density, finding that the maximal value of the\nKolmogorov-Sinai entropy is attained at infinite temperatures. Moreover, we\nrevisit the dynamical freezing of relaxation to equilibrium, occurring when\nlarge localized states (discrete breathers) are superposed to a generic\nfinite-temperature background. We show that the localized excitations induce a\nnumber of very small, yet not vanishing, Lyapunov exponents, which signal the\npresence of extremely long characteristic time-scales. We widen our analysis by\ncomputing the related Lyapunov covariant vectors, to investigate the\ninteraction of a single breather with the various degrees of freedom.", + "category": "nlin_CD" + }, + { + "text": "Quasiperiodic graphs at the onset of chaos: We examine the connectivity fluctuations across networks obtained when the\nhorizontal visibility (HV) algorithm is used on trajectories generated by\nnonlinear circle maps at the quasiperiodic transition to chaos. The resultant\nHV graph is highly anomalous as the degrees fluctuate at all scales with\namplitude that increases with the size of the network. We determine families of\nPesin-like identities between entropy growth rates and generalized\ngraph-theoretical Lyapunov exponents. An irrational winding number with pure\nperiodic continued fraction characterizes each family. We illustrate our\nresults for the so-called golden, silver and bronze numbers.", + "category": "nlin_CD" + }, + { + "text": "Coexisting solutions and their neighbourhood in the dynamical system\n describing second-order optical processes: Coexisting periodic solutions of a dynamical system describing nonlinear\noptical processes of the second-order are studied. The analytical results\nconcern both the simplified autonomous model and the extended nonautonomous\nmodel, including the pump and damping mechanism. The nonlinearity in the\ncoexisting solutions of the autonomous system is in concealed frequencies\ndepending on the initial conditions. In the solutions of the nonautonomous\nsystem the nonlinearity is convoluted in amplitudes. The neighbourhood of\nperiodic solutions is studied numerically, mainly in phase portraits. As a\nresult of disturbance, for example detuning, the periodic solutions are shown\nto escape to other states, periodic, quasiperiodic (beats) or chaotic. The\nchaotic behavior is indicated by the Lypunov exponents. We also investigate\nselected aspects of synchronization (unidirectional or mutual) of two identical\nsystems being in two different coexisting states. The effects of quenching of\noscillations are shown. In the autonomous system the quenching is caused by a\nchange in frequency, whereas in the nonautonomous one by a change in amplitude.\nThe quenching seems very promising for design of some advanced signal\nprocessing.", + "category": "nlin_CD" + }, + { + "text": "Fast chemical reaction in two-dimensional Navier-Stokes flow: Initial\n regime: This paper studies an infinitely fast bimolecular chemical reaction in a\ntwo-dimensional bi-periodic Navier-Stokes flow. The reactants in stoichiometric\nquantities are initially segregated by infinite gradients. The focus is placed\non the initial stage of the reaction characterized by a well-defined one\ndimensional material contact line between the reactants. Particular attention\nis given to the effect of the diffusion of the reactants. This study is an\nidealized framework for isentropic mixing in the lower stratosphere and is\nmotivated by the need to better understand the effect of resolutionon\nstratospheric chemistry in Climate-Chemistry Models.\n Adopting a Lagrangian stretching theory approach, we relate theoretically the\nensemble mean of the length of the contact line, of the gradients along it and\nof the modulus of the rate of decrease of the space averaged reactant\nconcentrations (here called the chemical speed) to the joint statistics of the\nfinite time Lyapunov exponent with two equivalent times. The inverse of the\nLyapunov exponent measures the stretching time scale of a Lagrangian parcel on\na chaotic orbit up to a finite time t, while the first equivalent time measures\nit in the recent past before t and the second equivalent time in the early part\nof the trajectory. We show that the chemical speed scales like the square root\nof the diffusion and that its time evolution is determined by rare large events\nin the finite time Lyapunov exponent distribution. The case of smooth initial\ngradients is also discussed. The theoretical results are tested with an\nensemble of direct numerical simulations (DNS) using a pseudospectral model.", + "category": "nlin_CD" + }, + { + "text": "Non-stationary resonance dynamics of the harmonically forced pendulum: The stationary and highly non-stationary resonant dynamics of the\nharmonically forced pendulum are described in the framework of a semi-inverse\nprocedure combined with the Limiting Phase Trajectory concept. This procedure,\nimplying only existence of slow time scale, permits to avoid any restriction on\nthe oscillation amplitudes. The main results relating to the dynamical\nbifurcation thresholds are represented in a closed form. The small parameter\ndefining the separation of the time scales is naturally identified in the\nanalytical procedure. Considering the pendulum frequency as the control\nparameter we reveal two qualitative transitions. One of them corresponding to\nstationary instability with formation of two additional stationary states, the\nother, associated with the most intense energy drawing from the source, at\nwhich the amplitude of pendulum oscillations abruptly grows.\n Analytical predictions of both bifurcations are verified by numerical\nintegration of original equation.\n It is also shown that occurrence of chaotic domains may be strongly connected\nwith the second transition.", + "category": "nlin_CD" + }, + { + "text": "Hidden transient chaotic attractors of Rabinovich-Fabrikant system: In [1], it is shown that the Rabinovich-Fabrikant (RF) system admits\nself-excited and hidden chaotic attractors. In this paper, we further show that\nthe RF system also admits a pair of symmetric transient hidden chaotic\nattractors. We reveal more extremely rich dynamics of this system, such as a\nnew kind of \"virtual saddles\".", + "category": "nlin_CD" + }, + { + "text": "Chaos from Symmetry: Navier Stokes equations, Beltrami fields and the\n Universal Classifying Crystallographic Group: The core of this paper is a novel group-theoretical approach, initiated in\n2015 by one of the present authors in collaboration with Alexander Sorin, which\nallows for a more systematic classification and algorithmic construction of\nBeltrami flows on torii $\\mathbb{R}^3/\\Lambda$ where $\\Lambda$ is a\ncrystallographic lattice. The new hydro-theory, based on the idea of a\nUniversal Classifying Group $\\mathfrak{UG}_\\Lambda$, is here revised. We\nconstruct the so far missing $\\mathfrak{UG}_{\\Lambda_{Hex}}$ for the hexagonal\nlattice. Mastering the cubic and hexagonal instances, we can cover all cases.\nThe classification relation between Beltrami Flows and contact structures is\nenlightened. The most promising research direction opened by the present work\nstreams from the fact that the Fourier series expansion of a generic\nNavier-Stokes solution can be regrouped into an infinite sum of contributions\n$\\mathbf{W}_r$, each associated with a spherical layer of quantized radius $r$\nin the momentum lattice and consisting of a superposition of a Beltrami and an\nanti-Beltrami field, with an analogous decomposition into irreps of the group\n$\\mathfrak{UG}_\\Lambda$ that are variously repeated on higher layers. This\ncrucial property enables the construction of generic Fourier series with\nprescribed hidden symmetries as candidate solutions of the NS equations.\n As a further result of this research programme a complete and versatile\nsystem of MATHEMATICA Codes named \\textbf{AlmafluidaNSPsystem} has been\nconstructed and is now available through the site of the Wolfram Community. The\nmain message streaming from our constructions is that the more symmetric the\nBeltrami Flow the highest is the probability of the onset of chaotic\ntrajectories.", + "category": "nlin_CD" + }, + { + "text": "Inferring Network Topology from Complex Dynamics: Inferring network topology from dynamical observations is a fundamental\nproblem pervading research on complex systems. Here, we present a simple,\ndirect method to infer the structural connection topology of a network, given\nan observation of one collective dynamical trajectory. The general theoretical\nframework is applicable to arbitrary network dynamical systems described by\nordinary differential equations. No interference (external driving) is required\nand the type of dynamics is not restricted in any way. In particular, the\nobserved dynamics may be arbitrarily complex; stationary, invariant or\ntransient; synchronous or asynchronous and chaotic or periodic. Presupposing a\nknowledge of the functional form of the dynamical units and of the coupling\nfunctions between them, we present an analytical solution to the inverse\nproblem of finding the network topology. Robust reconstruction is achieved in\nany sufficiently long generic observation of the system. We extend our method\nto simultaneously reconstruct both the entire network topology and all\nparameters appearing linear in the system's equations of motion. Reconstruction\nof network topology and system parameters is viable even in the presence of\nsubstantial external noise.", + "category": "nlin_CD" + }, + { + "text": "Experimental perspectives for systems based on long-range interactions: The possibility of observing phenomena peculiar to long-range interactions,\nand more specifically in the so-called Quasi-Stationary State (QSS) regime is\ninvestigated within the framework of two devices, namely the Free-Electron\nLaser (FEL) and the Collective Atomic Recoil Laser (CARL). The QSS dynamics has\nbeen mostly studied using the Hamiltonian Mean-Field (HMF) toy model,\ndemonstrating in particular the presence of first versus second order phase\ntransitions from magnetized to unmagnetized regimes in the case of HMF. Here,\nwe give evidence of the strong connections between the HMF model and the\ndynamics of the two mentioned devices, and we discuss the perspectives to\nobserve some specific QSS features experimentally. In particular, a dynamical\nanalog of the phase transition is present in the FEL and in the CARL in its\nconservative regime. Regarding the dissipative CARL, a formal link is\nestablished with the HMF model. For both FEL and CARL, calculations are\nperformed with reference to existing experimental devices, namely the\nFERMI@Elettra FEL under construction at Sincrotrone Trieste (Italy) and the\nCARL system at LENS in Florence (Italy).", + "category": "nlin_CD" + }, + { + "text": "Basin of Attraction Determines Hysteresis in Explosive Synchronization: Spontaneous explosive emergent behavior takes place in heterogeneous networks\nwhen the frequencies of the nodes are positively correlated to the node degree.\nA central feature of such explosive transitions is a hysteretic behavior at the\ntransition to synchronization. We unravel the underlying mechanisms and show\nthat the dynamical origin of the hysteresis is a change of basin of attraction\nof the synchronization state. Our findings hold for heterogeneous networks with\nstar graph motifs such as scale free networks, and hence reveal how microscopic\nnetwork parameters such as node degree and frequency affect the global network\nproperties and can be used for network design and control.", + "category": "nlin_CD" + }, + { + "text": "A Resolution of the Puzzle of the Posi-nega Switch Mechanism in the\n Globally Coupled Map Lattice: We revisit the globally coupled map lattice (GCML). We use a family of\nuniversal 'curves of balance' between two conflicting tendencies in the model-\nthe randomness in each map and the coherence due to an averaging interaction -\nand we locate the posi-nega switch region in the parameter space of the model.\nWe clarify the mechanism of a basic posi-nega switch in the two-cluster regime,\nwhich guarantees no mixing of maps across their mean field in the chaotic\ntransient process. We stress that a special attention must be paid for\nrounding-off errors; otherwise, there fatally occurs an artifitial numerical\ndegeneracy of maps in a cluster due to a highly negative Lyapunov exponent of\nthe clustor attractor.", + "category": "nlin_CD" + }, + { + "text": "Robust chaos in autonomous time-delay system: We consider an autonomous system constructed as modification of the logistic\ndifferential equation with delay that generates successive trains of\noscillations with phases evolving according to chaotic maps. The system\ncontains two feedback loops characterized by two generally distinct retarding\ntime parameters. In the case of their equality, chaotic dynamics is associated\nwith the Smale-Williams attractor that corresponds to the double-expanding\ncircle map for the phases of the carrier of the oscillatory trains.\nAlternatively, at appropriately chosen two different delays attractor is close\nto torus with Anosov dynamics on it as the phases are governed by the Fibonacci\nmap. In both cases the attractors manifest robustness (absence of regularity\nwindows under variation of parameters) and presumably relate to the class of\nstructurally stable hyperbolic attractors.", + "category": "nlin_CD" + }, + { + "text": "Broadband chaos generated by an opto-electronic oscillator: We study an opto-electronic time-delay oscillator that displays high-speed\nchaotic behavior with a flat, broad power spectrum. The chaotic state coexists\nwith a linearly-stable fixed point, which, when subjected to a finite-amplitude\nperturbation, loses stability initially via a periodic train of ultrafast\npulses. We derive an approximate map that does an excellent job of capturing\nthe observed instability. The oscillator provides a simple device for\nfundamental studies of time-delay dynamical systems and can be used as a\nbuilding block for ultra-wide-band sensor networks.", + "category": "nlin_CD" + }, + { + "text": "Geometrical origin of chaoticity in the bouncing ball billiard: We present a study of the chaotic behavior of the bouncing ball billiard. The\nwork is realised on the purpose of finding at least certain causes of\nseparation of the neighbouring trajectories. Having in view the geometrical\nconstruction of the system, we report a clear origin of chaoticity of the\nbouncing ball billiard. By this we claim that in case when the floor is made of\narc of circles - in a certain interval of frequencies - a lower bound for the\nmaximal Ljapunov can be evaluated by sem-ianalitical techniques.", + "category": "nlin_CD" + }, + { + "text": "Symmetry in the equations of nonlinear dynamics of semiconductor laser\n subject to delayed optical feedback: Delayed feedback laser dynamics is described by means of Lang-Kobayashi\nequation model. Since a lot of initial states asymptotically approach to\nperiodic attractor in the phase space, only periodic steady-state regimes have\nbeen studied here. Lyapunov transformation allows us to reduce problem to the\ndifferential equation of the first order whereas the spectrum of laser\noscillation is governed by the appropriate eigen value problem. Using the\nsymmetry proves that there is the dual laser system with anticipated feedback\nwhich has the same dynamic characteristics as laser with delayed feedback.", + "category": "nlin_CD" + }, + { + "text": "Pinning and depinning of a classic quasi-one-dimensional Wigner crystal\n in the presence of a constriction: We studied the dynamics of a quasi-one-dimensional chain-like system of\ncharged particles at low temperature, interacting through a screened Coulomb\npotential in the presence of a local constriction. The response of the system\nwhen an external electric field is applied was investigated. We performed\nLangevin molecular dynamics simulations for different values of the driving\nforce and for different temperatures. We found that the friction together with\nthe constriction pins the particles up to a critical value of the driving\nforce. The system can depin \\emph{elastically} or \\emph{quasi-elastically}\ndepending on the strength of the constriction. The elastic (quasi-elastic)\ndepinning is characterized by a critical exponent $\\beta\\sim0.66$\n($\\beta\\sim0.95$). The dc conductivity is zero in the pinned regime, it has\nnon-ohmic characteristics after the activation of the motion and then it is\nconstant. Furthermore, the dependence of the conductivity with temperature and\nstrength of the constriction was investigated in detail. We found interesting\ndifferences between the single and the multi-chain regimes as the temperature\nis increased.", + "category": "nlin_CD" + }, + { + "text": "From Turing patterns to chimera states in the 2D Brusselator model: The Brusselator has been used as a prototype model for autocatalytic\nreactions, and in particular for the Belouzov- Zhabotinsky reaction. When\ncoupled at the diffusive limit, the Brusselator undergoes a Turing bifurcation\nresulting in the formation of classical Turing patterns, such as spots, stripes\nand spirals in 2 spatial dimensions. In the present study we use generic\nnonlocally coupled Brusselators and show that in the limit of the coupling\nrange R->1 (diffusive limit), the classical Turing patterns are recovered,\nwhile for intermediate coupling ranges and appropriate parameter values chimera\nstates are produced. This study demonstrates how the parameters of a typical\nnonlinear oscillator can be tuned so that the coupled system passes from\nspatially stable Turing structures to dynamical spatiotemporal chimera states.", + "category": "nlin_CD" + }, + { + "text": "A new approach to simulating stochastic delayed systems: In this paper we present a new method for deriving It\\^{o} stochastic delay\ndifferential equations (SDDEs) from delayed chemical master equations (DCMEs).\nConsidering alternative formulations of SDDEs that can be derived from the same\nDCME, we prove that they are equivalent both in distribution, and in sample\npaths they produce. This allows us to formulate an algorithmic approach to\nderiving equivalent It\\^{o} SDDEs with a smaller number of noise variables,\nwhich increases the computational speed of simulating stochastic delayed\nsystems. The new method is illustrated on a simple model of two interacting\nspecies, and it shows excellent agreement with the results of direct stochastic\nsimulations, while also demonstrating a much superior speed of performance.", + "category": "nlin_CD" + }, + { + "text": "Experimental and numerical investigation of the reflection coefficient\n and the distributions of Wigner's reaction matrix for irregular graphs with\n absorption: We present the results of experimental and numerical study of the\ndistribution of the reflection coefficient P(R) and the distributions of the\nimaginary P(v) and the real P(u) parts of the Wigner's reaction K matrix for\nirregular fully connected hexagon networks (graphs) in the presence of strong\nabsorption. In the experiment we used microwave networks, which were built of\ncoaxial cables and attenuators connected by joints. In the numerical\ncalculations experimental networks were described by quantum fully connected\nhexagon graphs. The presence of absorption introduced by attenuators was\nmodelled by optical potentials. The distribution of the reflection coefficient\nP(R) and the distributions of the reaction K matrix were obtained from the\nmeasurements and numerical calculations of the scattering matrix S of the\nnetworks and graphs, respectively. We show that the experimental and numerical\nresults are in good agreement with the exact analytic ones obtained within the\nframework of random matrix theory (RMT).", + "category": "nlin_CD" + }, + { + "text": "A mechanical mode-stirred reverberation chamber with chaotic geometry: A previous research on multivariate approach to the calculation of\nreverberation chamber correlation matrices is used to calculate the number of\nindependent positions in a mode-stirred reverberation chamber. Anomalies and\ncounterintuitive behavior are observed in terms of number of correlated matrix\nelements with respect to increasing frequency. This is ascribed to the regular\ngeometry forming the baseline cavity (screened room) of a reverberation\nchamber, responsible for localizing energy and preserving regular modes\n(bouncing ball modes). Smooth wall deformations are introduced in order to\ncreate underlying Lyapunov instability of rays and then destroy survived\nregular modes. Numerical full-wave simulations are performed for a\nreverberation chamber with corner hemispheres and (off-)center wall spherical\ncaps. Field sampling is performed by moving a mechanical carousel stirrer. It\nis found that wave-chaos inspired baseline geometries improve chamber\nperformances in terms of lowest usable frequencies and number of independent\ncavity realizations of mechanical stirrers.", + "category": "nlin_CD" + }, + { + "text": "Solution of linearized Fokker - Planck equation for incompressible fluid: In this work we construct algebraic equation for elements of spectrum of\nlinearized Fokker - Planck differential operator for incompressible fluid. We\ncalculate roots of this equation using simple numeric method. For all these\nroots real part is positive, that is corresponding solutions are damping.\nEigenfunctions of linearized Fokker - Planck differential operator for\nincompressible fluid are expressed as linear combinations of eigenfunctions of\nusual Fokker - Planck differential operator. Poisson's equation for pressure is\nderived from incompressibility condition. It is stated, that the pressure could\nbe totally eliminated from dynamics equations. The Cauchy problem setup and\nsolution method is presented. The role of zero pressure solutions as\neigenfunctions for confluent eigenvalues is emphasized.", + "category": "nlin_CD" + }, + { + "text": "Unstable periodic orbits in a chaotic meandering jet flow: We study the origin and bifurcations of typical classes of unstable periodic\norbits in a jet flow that was introduced before as a kinematic model of chaotic\nadvection, transport and mixing of passive scalars in meandering oceanic and\natmospheric currents. A method to detect and locate the unstable periodic\norbits and classify them by the origin and bifurcations is developed. We\nconsider in detail period-1 and period-4 orbits playing an important role in\nchaotic advection. We introduce five classes of period-4 orbits: western and\neastern ballistic ones, whose origin is associated with ballistic resonances of\nthe fourth order, rotational ones, associated with rotational resonances of the\nsecond and fourth orders, and rotational-ballistic ones associated with a\nrotational-ballistic resonance. It is a new kind of nonlinear resonances that\nmay occur in chaotic flow with jets and/or circulation cells. Varying the\nperturbation amplitude, we track out the origin and bifurcations of the orbits\nfor each class.", + "category": "nlin_CD" + }, + { + "text": "Random Matrix Spectra as a Time Series: Spectra of ordered eigenvalues of finite Random Matrices are interpreted as a\ntime series. Dataadaptive techniques from signal analysis are applied to\ndecompose the spectrum in clearly differentiated trend and fluctuation modes,\navoiding possible artifacts introduced by standard unfolding techniques. The\nfluctuation modes are scale invariant and follow different power laws for\nPoisson and Gaussian ensembles, which already during the unfolding allows to\ndistinguish the two cases.", + "category": "nlin_CD" + }, + { + "text": "Unified Model, and Novel Reverse Recovery Nonlinearities, of the Driven\n Diode Resonator: We study the origins of period doubling and chaos in the driven series\nresistor-inductor-varactor diode (RLD) nonlinear resonant circuit. We find that\nresonators driven at frequencies much higher than the diode reverse recovery\nrate do not show period doubling, and that models of chaos based on the\nnonlinear capacitance of the varactor diode display a reverse-recovery-like\neffect, and this effect strongly resembles reverse recovery of real diodes. We\nfind for the first time that in addition to the known dependence of the reverse\nrecovery time on past current maxima, there are also important nonlinear\ndependencies on pulse frequency, duty-cycle, and DC voltage bias. Similar\nnonlinearities are present in the nonlinear capacitance models of these diodes.\nWe conclude that a history-dependent and nonlinear reverse recovery time is an\nessential ingredient for chaotic behavior of this circuit, and demonstrate for\nthe first time that all major competing models have this effect, either\nexplicitly or implicitly. Besides unifying the two major models of RLD chaos,\nour work reveals that the nonlinearities of the reverse recovery time must be\nincluded for a complete understanding of period doubling and chaos in this\ncircuit.", + "category": "nlin_CD" + }, + { + "text": "Arnold diffusion of charged particles in ABC magnetic fields: We prove the existence of diffusing solutions in the motion of a charged\nparticle in the presence of an ABC magnetic field. The equations of motion are\nmodeled by a 3DOF Hamiltonian system depending on two parameters. For small\nvalues of these parameters, we obtain a normally hyperbolic invariant manifold\nand we apply the so-called geometric methods for a priori unstable systems\ndeveloped by A. Delshams, R. de la Llave, and T.M. Seara. We characterize\nexplicitly sufficient conditions for the existence of a transition chain of\ninvariant tori having heteroclinic connections, thus obtaining global\ninstability (Arnold diffusion). We also check the obtained conditions in a\ncomputer assisted proof. ABC magnetic fields are the simplest force-free type\nsolutions of the magnetohydrodynamics equations with periodic boundary\nconditions, so our results are of potential interest in the study of the motion\nof plasma charged particles in a tokamak.", + "category": "nlin_CD" + }, + { + "text": "Symmetry broken states in an ensemble of globally coupled pendulums: We consider the rotational dynamics in an ensemble of globally coupled\nidentical pendulums. This model is essentially a generalization of the standard\nKuramoto model, which takes into account the inertia and the intrinsic\nnonlinearity of the community elements. There exists the wide variety of\nin-phase and out-of-phase regimes. Many of these states appear due to broken\nsymmetry. In the case of small dissipation our theoretical analysis allows one\nto find the boundaries of the instability domain of in-phase rotational mode\nfor ensembles with arbitrary number of pendulums, describe all arising\nout-of-phase rotation modes and study in detail their stability. For the system\nof three elements parameter sets corresponding to the unstable in-phase\nrotations we find a number of out-of-phase regimes and investigate their\nstability and bifurcations both analytically and numerically. As a result, we\nobtain a sufficiently detailed picture of the symmetry breaking and existence\nof various regular and chaotic states.", + "category": "nlin_CD" + }, + { + "text": "A moment-equation-copula-closure method for nonlinear vibrational\n systems subjected to correlated noise: We develop a moment equation closure minimization method for the inexpensive\napproximation of the steady state statistical structure of nonlinear systems\nwhose potential functions have bimodal shapes and which are subjected to\ncorrelated excitations. Our approach relies on the derivation of moment\nequations that describe the dynamics governing the two-time statistics. These\nare combined with a non-Gaussian pdf representation for the joint\nresponse-excitation statistics that has i) single time statistical structure\nconsistent with the analytical solutions of the Fokker-Planck equation, and ii)\ntwo-time statistical structure with Gaussian characteristics. Through the\nadopted pdf representation, we derive a closure scheme which we formulate in\nterms of a consistency condition involving the second order statistics of the\nresponse, the closure constraint. A similar condition, the dynamics constraint,\nis also derived directly through the moment equations. These two constraints\nare formulated as a low-dimensional minimization problem with respect to\nunknown parameters of the representation, the minimization of which imposes an\ninterplay between the dynamics and the adopted closure. The new method allows\nfor the semi-analytical representation of the two-time, non-Gaussian structure\nof the solution as well as the joint statistical structure of the\nresponse-excitation over different time instants. We demonstrate its\neffectiveness through the application on bistable nonlinear\nsingle-degree-of-freedom energy harvesters with mechanical and electromagnetic\ndamping, and we show that the results compare favorably with direct Monte-Carlo\nSimulations.", + "category": "nlin_CD" + }, + { + "text": "A taxonomy for generalized synchronization between flat-coupled systems: Generalized synchronization is plausibly the most complex form of\nsynchronization. Previous studies have revealed the existence of weak or strong\nforms of generalized synchronization depending on the multi- or mono-valued\nnature of the mapping between the attractors of two unidirectionally-coupled\nsystems. Generalized synchronization is here obtained by coupling two systems\nwith a flat control law. Here, we demonstrate that the corresponding\nfirst-return maps can be topologically conjugate in some cases. Conversely, the\nresponse map can foliated while the drive map is not. We describe the\ncorresponding types of generalized synchronization, explicitly focusing on the\ninfluence of the coupling strength when significantly different dimensions or\ndissipation properties characterize the coupled systems. A taxonomy of\ngeneralized synchronization based on these properties is proposed.", + "category": "nlin_CD" + }, + { + "text": "Symbolic Synchronization and the Detection of Global Properties of\n Coupled Dynamics from Local Information: We study coupled dynamics on networks using symbolic dynamics. The symbolic\ndynamics is defined by dividing the state space into a small number of regions\n(typically 2), and considering the relative frequencies of the transitions\nbetween those regions. It turns out that the global qualitative properties of\nthe coupled dynamics can be classified into three different phases based on the\nsynchronization of the variables and the homogeneity of the symbolic dynamics.\nOf particular interest is the {\\it homogeneous unsynchronized phase} where the\ncoupled dynamics is in a chaotic unsynchronized state, but exhibits (almost)\nidentical symbolic dynamics at all the nodes in the network. We refer to this\ndynamical behaviour as {\\it symbolic synchronization}. In this phase, the local\nsymbolic dynamics of any arbitrarily selected node reflects global properties\nof the coupled dynamics, such as qualitative behaviour of the largest Lyapunov\nexponent and phase synchronization. This phase depends mainly on the network\narchitecture, and only to a smaller extent on the local chaotic dynamical\nfunction. We present results for two model dynamics, iterations of the\none-dimensional logistic map and the two-dimensional H\\'enon map, as local\ndynamical function.", + "category": "nlin_CD" + }, + { + "text": "Chaotic dynamics with Maxima: We present an introduction to the study of chaos in discrete and continuous\ndynamical systems using the CAS Maxima. These notes are intended to cover the\nstandard topics and techniques: discrete and continuous logistic equation to\nmodel growth population, staircase plots, bifurcation diagrams and chaos\ntransition, nonlinear continuous dynamics (Lorentz system and Duffing\noscillator), Lyapunov exponents, Poincar\\'e sections, fractal dimension and\nstrange attractors. The distinctive feature here is the use of free software\nwith just one ingredient: the CAS Maxima. It is cross-platform and have\nextensive on-line documentation.", + "category": "nlin_CD" + }, + { + "text": "Accurately Estimating the State of a Geophysical System with Sparse\n Observations: Predicting the Weather: Utilizing the information in observations of a complex system to make\naccurate predictions through a quantitative model when observations are\ncompleted at time $T$, requires an accurate estimate of the full state of the\nmodel at time $T$.\n When the number of measurements $L$ at each observation time within the\nobservation window is larger than a sufficient minimum value $L_s$, the\nimpediments in the estimation procedure are removed. As the number of available\nobservations is typically such that $L \\ll L_s$, additional information from\nthe observations must be presented to the model.\n We show how, using the time delays of the measurements at each observation\ntime, one can augment the information transferred from the data to the model,\nremoving the impediments to accurate estimation and permitting dependable\nprediction. We do this in a core geophysical fluid dynamics model, the shallow\nwater equations, at the heart of numerical weather prediction. The method is\nquite general, however, and can be utilized in the analysis of a broad spectrum\nof complex systems where measurements are sparse. When the model of the complex\nsystem has errors, the method still enables accurate estimation of the state of\nthe model and thus evaluation of the model errors in a manner separated from\nuncertainties in the data assimilation procedure.", + "category": "nlin_CD" + }, + { + "text": "Chaotic Phenomenon in Nonlinear Gyrotropic Medium: Nonlinear gyrotropic medium is a medium, whose natural optical activity\ndepends on the intensity of the incident light wave. The Kuhn's model is used\nto study nonlinear gyrotropic medium with great success. The Kuhn's model\npresents itself a model of nonlinear coupled oscillators. This article is\ndevoted to the study of the Kuhn's nonlinear model. In the first paragraph of\nthe paper we study classical dynamics in case of weak as well as strong\nnonlinearity. In case of week nonlinearity we have obtained the analytical\nsolutions, which are in good agreement with the numerical solutions. In case of\nstrong nonlinearity we have determined the values of those parameters for which\nchaos is formed in the system under study. The second paragraph of the paper\nrefers to the question of the Kuhn's model integrability. It is shown, that at\nthe certain values of the interaction potential this model is exactly\nintegrable and under certain conditions it is reduced to so-called universal\nHamiltonian. The third paragraph of the paper is devoted to quantum-mechanical\nconsideration. It shows the possibility of stochastic absorption of external\nfield energy by nonlinear gyrotropic medium. The last forth paragraph of the\npaper is devoted to generalization of the Kuhn's model for infinite chain of\ninteracting oscillators.", + "category": "nlin_CD" + }, + { + "text": "Theory of localization and resonance phenomena in the quantum kicked\n rotor: We present an analytic theory of quantum interference and Anderson\nlocalization in the quantum kicked rotor (QKR). The behavior of the system is\nknown to depend sensitively on the value of its effective Planck's constant\n$\\he$. We here show that for rational values of $\\he/(4\\pi)=p/q$, it bears\nsimilarity to a disordered metallic ring of circumference $q$ and threaded by\nan Aharonov-Bohm flux. Building on that correspondence, we obtain quantitative\nresults for the time--dependent behavior of the QKR kinetic energy, $E(\\tilde\nt)$ (this is an observable which sensitively probes the system's localization\nproperties). For values of $q$ smaller than the localization length $\\xi$, we\nobtain scaling $E(\\tilde t) \\sim \\Delta \\tilde t^2$, where $\\Delta=2\\pi/q$ is\nthe quasi--energy level spacing on the ring. This scaling is indicative of a\nlong time dynamics that is neither localized nor diffusive. For larger values\n$q\\gg \\xi$, the functions $E(\\tilde t)\\to \\xi^2$ saturates (up to exponentially\nsmall corrections $\\sim\\exp(-q/\\xi)$), thus reflecting essentially localized\nbehavior.", + "category": "nlin_CD" + }, + { + "text": "Chaos synchronization with coexisting global fields: We investigate the phenomenon of chaos synchronization in systems subject to\ncoexisting autonomous and external global fields by employing a simple model of\ncoupled maps. Two states of chaos synchronization are found: (i) complete\nsynchronization, where the maps synchronize among themselves and to the\nexternal field, and (ii) generalized or internal synchronization, where the\nmaps synchronize among themselves but not to the external global field. We show\nthat the stability conditions for both states can be achieved for a system of\nminimum size of two maps. We consider local maps possessing robust chaos and\ncharacterize the synchronization states on the space of parameters of the\nsystem. The state of generalized synchronization of chaos arises even the drive\nand the local maps have the same functional form. This behavior is similar to\nthe process of spontaneous ordering against an external field found in\nnonequilibrium systems.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of rolling disk: In the paper we present the qualitative analysis of rolling motion without\nslipping of a homogeneous round disk on a horisontal plane. The problem was\nstudied by S.A. Chaplygin, P. Appel and D. Korteweg who showed its\nintegrability. The behavior of the point of contact on a plane is investigated\nand conditions under which its trajectory is finit are obtained. The\nbifurcation diagrams are constructed.", + "category": "nlin_CD" + }, + { + "text": "Low dimensional behavior in three-dimensional coupled map lattices: The analysis of one-, two-, and three-dimensional coupled map lattices is\nhere developed under a statistical and dynamical perspective. We show that the\nthree-dimensional CML exhibits low dimensional behavior with long range\ncorrelation and the power spectrum follows $1/f$ noise. This approach leads to\nan integrated understanding of the most important properties of these universal\nmodels of spatiotemporal chaos. We perform a complete time series analysis of\nthe model and investigate the dependence of the signal properties by change of\ndimension.", + "category": "nlin_CD" + }, + { + "text": "Duffing-type equations: singular points of amplitude profiles and\n bifurcations: We study the Duffing equation and its generalizations with polynomial\nnonlinearities. Recently, we have demonstrated that metamorphoses of the\namplitude response curves, computed by asymptotic methods in implicit form as\n$F\\left( \\Omega ,\\ A\\right) =0$, permit prediction of qualitative changes of\ndynamics occurring at singular points of the implicit curve $F\\left(\\Omega ,\\\nA\\right) =0$. In the present work we determine a global structure of singular\npoints of the amplitude profiles computing bifurcation sets, i.e. sets\ncontaining all points in the parameter space for which the amplitude profile\nhas a singular point. We connect our work with independent research on\ntangential points on amplitude profiles, associated with jump phenomena,\ncharacteristic for the Duffing equation. We also show that our techniques can\nbe applied to solutions of form $\\Omega _{\\pm }=f_{\\pm }\\left( A\\right) $,\nobtained within other asymptotic approaches.", + "category": "nlin_CD" + }, + { + "text": "Characteristic times for the Fermi-Ulam Model: The mean Poincarr\\'e recurrence time as well as the Lyapunov time are\nmeasured for the Fermi-Ulam model. We confirm the mean recurrence time is\ndependent on the size of the window chosen in the phase space to where\nparticles are allowed to recur. The fractal dimension of the region is\ndetermined by the slope of the recurrence time against the size of the window\nand two numerical values were measured: (i) $\\mu$ = 1 confirming normal\ndiffusion for chaotic regions far from periodic domains and; (ii) $\\mu$ = 2\nleading to anomalous diffusion measured near periodic regions, a signature of\nlocal trapping of an ensemble of particles. The Lyapunov time is measured over\ndifferent domains in the phase space through a direct determination of the\nLyapunov exponent, indeed being defined as its inverse.", + "category": "nlin_CD" + }, + { + "text": "Generalized Chaotic Synchronizationin Coupled Ginzburg-Landau Equations: Generalized synchronization is analyzed in unidirectionally coupled\noscillatory systems exhibiting spatiotemporal chaotic behavior described by\nGinzburg-Landau equations. Several types of coupling betweenthe systems are\nanalyzed. The largest spatial Lyapunov exponent is proposed as a new\ncharacteristic of the state of a distributed system, and its calculation is\ndescribed for a distributed oscillatory system. Partial generalized\nsynchronization is introduced as a new type of chaotic synchronization in\nspatially nonuniform distributed systems. The physical mechanisms responsible\nfor the onset of generalized chaotic synchronization in spatially distributed\noscillatory systems are elucidated. It is shown that the onset of generalized\nchaotic synchronization is described by a modified Ginzburg-Landau equation\nwith additional dissipation irrespective of the type of coupling. The effect of\nnoise on the onset of a generalized synchronization regime in coupled\ndistributed systems is analyzed.", + "category": "nlin_CD" + }, + { + "text": "Naimark-Sacker Bifurcations in Linearly Coupled Quadratic Maps: We report exact analytical expressions locating the $0\\to1$, $1\\to2$ and\n$2\\to4$ bifurcation curves for a prototypical system of two linearly coupled\nquadratic maps. Of interest is the precise location of the parameter sets where\nNaimark-Sacker bifurcations occur, starting from a non-diagonal period-2 orbit.\nThis result is the key to understand the onset of synchronization in networks\nof quadratic maps.", + "category": "nlin_CD" + }, + { + "text": "Scaling Analysis and Evolution Equation of the North Atlantic\n Oscillation Index Fluctuations: The North Atlantic Oscillation (NAO) monthly index is studied from 1825 till\n2002 in order to identify the scaling ranges of its fluctuations upon different\ndelay times and to find out whether or not it can be regarded as a Markov\nprocess. A Hurst rescaled range analysis and a detrended fluctuation analysis\nboth indicate the existence of weakly persistent long range time correlations\nfor the whole scaling range and time span hereby studied. Such correlations are\nsimilar to Brownian fluctuations. The Fokker-Planck equation is derived and\nKramers-Moyal coefficients estimated from the data. They are interpreted in\nterms of a drift and a diffusion coefficient as in fluid mechanics. All partial\ndistribution functions of the NAO monthly index fluctuations have a form close\nto a Gaussian, for all time lags, in agreement with the findings of the scaling\nanalyses. This indicates the lack of predictive power of the present NAO\nmonthly index. Yet there are some deviations for large (and thus rare) events.\nWhence suggestions for other measurements are made if some improved\npredictability of the weather/climate in the North Atlantic is of interest. The\nsubsequent Langevin equation of the NAO signal fluctuations is explicitly\nwritten in terms of the diffusion and drift parameters, and a characteristic\ntime scale for these is given in appendix.", + "category": "nlin_CD" + }, + { + "text": "Exact geometric theory of dendronized polymer dynamics: Dendronized polymers consist of an elastic backbone with a set of iterated\nbranch structures (dendrimers)attached at every base point of the backbone. The\nconformations of such molecules depend on the elastic deformation of the\nbackbone and the branches, as well as on nonlocal (e.g., electrostatic, or\nLennard-Jones) interactions between the elementary molecular units comprising\nthe dendrimers and/or backbone. We develop a geometrically exact theory for the\ndynamics of such polymers, taking into account both local (elastic) and\nnonlocal interactions. The theory is based on applying symmetry reduction of\nHamilton's principle for a Lagrangian defined on the tangent bundle of iterated\nsemidirect products of the rotation groups that represent the relative\norientations of the dendritic branches of the polymer. The resulting\nsymmetry-reduced equations of motion are written in conservative form.", + "category": "nlin_CD" + }, + { + "text": "Rich dynamics and anticontrol of extinction in a prey-predator system: This paper reveals some new and rich dynamics of a two-dimensional\nprey-predator system and to anticontrol the extinction of one of the species.\nFor a particular value of the bifurcation parameter, one of the system variable\ndynamics is going to extinct, while another remains chaotic. To prevent the\nextinction, a simple anticontrol algorithm is applied so that the system orbits\ncan escape from the vanishing trap. As the bifurcation parameter increases, the\nsystem presents quasiperiodic, stable, chaotic and also hyperchaotic orbits.\nSome of the chaotic attractors are Kaplan-Yorke type, in the sense that the sum\nof its Lyapunov exponents is positive. Also, atypically for undriven discrete\nsystems, it is numerically found that, for some small parameter ranges, the\nsystem seemingly presents strange nonchaotic attractors. It is shown both\nanalytically and by numerical simulations that the original system and the\nanticontrolled system undergo several Neimark-Sacker bifurcations. Beside the\nclassical numerical tools for analyzing chaotic systems, such as phase\nportraits, time series and power spectral density, the 0-1 test is used to\ndifferentiate regular attractors from chaotic attractors.", + "category": "nlin_CD" + }, + { + "text": "The bifurcations of the critical points and the role of the depth in a\n symmetric Caldera potential energy surface: In this work, we continue the study of the bifurcations of the critical\npoints in a symmetric Caldera potential energy surface. In particular, we study\nthe influence of the depth of the potential on the trajectory behavior before\nand after the bifurcations of the critical points. We observe two different\ntypes of trajectory behavior: dynamical matching and the non-existence of\ndynamical matching. Dynamical matching is a phenomenon that limits the way in\nwhich a trajectory can exit the Caldera based solely on how it enters the\nCaldera. Furthermore, we discuss two different types of symmetric Caldera\npotential energy surface and the transition from the one type to the other\nthrough the bifurcations of the critical points.", + "category": "nlin_CD" + }, + { + "text": "Chaotic motion of charged particles in toroidal magnetic configurations: We study the motion of a charged particle in a tokamak magnetic field and\ndiscuss its chaotic nature. Contrary to most of recent studies, we do not make\nany assumption on any constant of the motion and solve numerically the\ncyclotron gyration using Hamiltonian formalism. We take advantage of a\nsymplectic integrator allowing us to make long-time simulations. First\nconsidering an idealized magnetic configuration, we add a non generic\nperturbation corresponding to a magnetic ripple, breaking one of the invariant\nof the motion. Chaotic motion is then observed and opens questions about the\nlink between chaos of magnetic field lines and chaos of particle trajectories.\nSecond, we return to a axi-symmetric configuration and tune the safety factor\n(magnetic configuration) in order to recover chaotic motion. In this last\nsetting with two constants of the motion, the presence of chaos implies that no\nthird global constant exists, we highlight this fact by looking at variations\nof the first order of the magnetic moment in this chaotic setting. We are\nfacing a mixed phase space with both regular and chaotic regions and point out\nthe difficulties in performing a global reduction such as gyrokinetics.", + "category": "nlin_CD" + }, + { + "text": "Logarithmic periodicities in the bifurcations of type-I intermittent\n chaos: The critical relations for statistical properties on saddle-node bifurcations\nare shown to display undulating fine structure, in addition to their known\nsmooth dependence on the control parameter. A piecewise linear map with the\ntype-I intermittency is studied and a log-periodic dependence is numerically\nobtained for the average time between laminar events, the Lyapunov exponent and\nattractor moments. The origin of the oscillations is built in the natural\nprobabilistic measure of the map and can be traced back to the existence of\nlogarithmically distributed discrete values of the control parameter giving\nMarkov partition. Reinjection and noise effect dependences are discussed and\nindications are given on how the oscillations are potentially applicable to\ncomplement predictions made with the usual critical exponents, taken from data\nin critical phenomena.", + "category": "nlin_CD" + }, + { + "text": "Estimation of initial conditions from a scalar time series: We introduce a method to estimate the initial conditions of a mutivariable\ndynamical system from a scalar signal. The method is based on a modified\nmultidimensional Newton-Raphson method which includes the time evolution of the\nsystem. The method can estimate initial conditions of periodic and chaotic\nsystems and the required length of scalar signal is very small. Also, the\nmethod works even when the conditional Lyapunov exponent is positive. An\nimportant application of our method is that synchronization of two chaotic\nsignals using a scalar signal becomes trivial and instantaneous.", + "category": "nlin_CD" + }, + { + "text": "Pseudo resonance induced quasi-periodic behavior in stochastic threshold\n dynamics: Here we present a simple stochastic threshold model consisting of a\ndeterministic slowly decaying term and a fast stochastic noise term. The\nprocess shows a pseudo-resonance, in the sense that for small and large\nintensities of the noise the signal is irregular and the distribution of\nthreshold crossings is broad, while for a tuned intermediate value of noise\nintensity the signal becomes quasi-periodic and the distribution of threshold\ncrossings is narrow. The mechanism captured by the model might be relevant for\nexplaining apparent quasi-periodicity of observed climatic variations where no\ninternal or external periodicities can be identified.", + "category": "nlin_CD" + }, + { + "text": "Universality and Hysteresis in Slow Sweeping of Bifurcations: Bifurcations in dynamical systems are often studied experimentally and\nnumerically using a slow parameter sweep. Focusing on the cases of\nperiod-doubling and pitchfork bifurcations in maps, we show that the adiabatic\napproximation always breaks down sufficiently close to the bifurcation, so that\nthe upsweep and downsweep dynamics diverge from one another, disobeying\nstandard bifurcation theory. Nevertheless, we demonstrate universal upsweep and\ndownsweep trajectories for sufficiently slow sweep rates, revealing that the\nslow trajectories depend essentially on a structural asymmetry parameter, whose\neffect is negligible for the stationary dynamics. We obtain explicit asymptotic\nexpressions for the universal trajectories, and use them to calculate the area\nof the hysteresis loop enclosed between the upsweep and downsweep trajectories\nas a function of the asymmetry parameter and the sweep rate.", + "category": "nlin_CD" + }, + { + "text": "Sampling Chaotic Trajectories Quickly in Parallel: The parallel computational complexity of the quadratic map is studied. A\nparallel algorithm is described that generates typical pseudotrajectories of\nlength t in a time that scales as log t and increases slowly in the accuracy\ndemanded of the pseudotrajectory. Long pseudotrajectories are created in\nparallel by putting together many short pseudotrajectories; Monte Carlo\nprocedures are used to eliminate the discontinuities between these short\npseudotrajectories and then suitably randomize the resulting long\npseudotrajectory. Numerical simulations are presented that show the scaling\nproperties of the parallel algorithm. The existence of the fast parallel\nalgorithm provides a way to formalize the intuitive notion that chaotic systems\ndo not generate complex histories.", + "category": "nlin_CD" + }, + { + "text": "Hiding message in Delay Time: Encryption with Synchronized time-delayed\n systems: We propose a new communication scheme that uses time-delayed chaotic systems\nwith delay time modulation. In this method, the transmitter encodes a message\nas an additional modulation of the delay timeand then the receiver decodes the\nmessage by tracking the delay time.We demonstrate our communication scheme in a\nsystem of coupled logistic maps.Also we discuss the error of the transferred\nmessage due to an external noiseand present its correction method.", + "category": "nlin_CD" + }, + { + "text": "Non-Reversible Evolution of Quantum Chaotic System. Kinetic Description: Time dependent dynamics of the chaotic quantum-mechanical system has been\nstudied. Irreversibility of the dynamics is shown. It is shown, that being in\nthe initial moment in pure quantum-mechanical state, system makes irreversible\ntransition into mixed state. Original mechanism of mixed state formation is\noffered. Quantum kinetic equation is obtained. Growth of the entropy during the\nevolution process is estimated.", + "category": "nlin_CD" + }, + { + "text": "Classification and stability of simple homoclinic cycles in R^5: The paper presents a complete study of simple homoclinic cycles in R^5. We\nfind all symmetry groups Gamma such that a Gamma-equivariant dynamical system\nin R^5 can possess a simple homoclinic cycle. We introduce a classification of\nsimple homoclinic cycles in R^n based on the action of the system symmetry\ngroup. For systems in R^5, we list all classes of simple homoclinic cycles. For\neach class, we derive necessary and sufficient conditions for asymptotic\nstability and fragmentary asymptotic stability in terms of eigenvalues of\nlinearisation near the steady state involved in the cycle. For any action of\nthe groups Gamma which can give rise to a simple homoclinic cycle, we list\nclasses to which the respective homoclinic cycles belong, thus determining\nconditions for asymptotic stability of these cycles.", + "category": "nlin_CD" + }, + { + "text": "Determination of fractal dimensions of solar radio bursts: We present a dimension analysis of a set of solar type I storms and type IV\nevents with different kind of fine structures, recorded at the Trieste\nAstronomical Observatory. The signature of such types of solar radio events is\nhighly structured in time. However, periodicities are rather seldom, and linear\nmode theory can provide only limited interpretation of the data. Therefore, we\nperformed an analysis based on methods of the nonlinear dynamics theory.\nAdditionally to the commonly used correlation dimension, we also calculated\nlocal pointwise dimensions. This alternative approach is motivated by the fact\nthat astrophysical time series represent real-world systems, which cannot be\nkept in a controlled state and which are highly interconnected with their\nsurroundings. In such systems pure determinism is rather unlikely to be\nrealized, and therefore a characterization by invariants of the dynamics might\nprobably be inadequate. In fact, the outcome of the dimension analysis does not\ngive hints for low-dimensional determinism in the data, but we show that,\ncontrary to the correlation dimension method, local dimension estimations can\ngive physical insight into the events even in cases in which pure determinism\ncannot be established. In particular, in most of the analyzed radio events\nnonlinearity in the data is detected, and the local dimension analysis provides\na basis for a quantitative description of the time series, which can be used to\ncharacterize the complexity of the related physical system in a comparative and\nnon-invariant manner.", + "category": "nlin_CD" + }, + { + "text": "Bubbling in delay-coupled lasers: We theoretically study chaos synchronization of two lasers which are\ndelay-coupled via an active or a passive relay. While the lasers are\nsynchronized, their dynamics is identical to a single laser with delayed\nfeedback for a passive relay and identical to two delay-coupled lasers for an\nactive relay. Depending on the coupling parameters the system exhibits\nbubbling, i.e., noise-induced desynchronization, or on-off intermittency. We\nassociate the desynchronization dynamics in the coherence collapse and low\nfrequency fluctuation regimes with the transverse instability of some of the\ncompound cavity's antimodes. Finally, we demonstrate how, by using an active\nrelay, bubbling can be suppressed.", + "category": "nlin_CD" + }, + { + "text": "Synchronization transitions in globally coupled rotors in presence of\n noise and inertia: Exact results: We study a generic model of globally coupled rotors that includes the effects\nof noise, phase shift in the coupling, and distributions of moments of inertia\nand natural frequencies of oscillation. As particular cases, the setup includes\npreviously studied Sakaguchi-Kuramoto, Hamiltonian and Brownian mean-field, and\nTanaka-Lichtenberg-Oishi and Acebr\\'on-Bonilla-Spigler models. We derive an\nexact solution of the self-consistent equations for the order parameter in the\nstationary state, valid for arbitrary parameters in the dynamics, and\ndemonstrate nontrivial phase transitions to synchrony that include reentrant\nsynchronous regimes.", + "category": "nlin_CD" + }, + { + "text": "Optimal Tree for Both Synchronizability and Converging Time: It has been proved that the spanning tree from a given network has the\noptimal synchronizability, which means the index $R=\\lambda_{N}/\\lambda_{2}$\nreaches the minimum 1. Although the optimal synchronizability is corresponding\nto the minimal critical overall coupling strength to reach synchronization, it\ndoes not guarantee a shorter converging time from disorder initial\nconfiguration to synchronized state. In this letter, we find that it is the\ndepth of the tree that affects the converging time. In addition, we present a\nsimple and universal way to get such an effective oriented tree in a given\nnetwork to reduce the converging time significantly by minimizing the depth of\nthe tree. The shortest spanning tree has both the maximal synchronizability and\nefficiency.", + "category": "nlin_CD" + }, + { + "text": "Reply to a Comment by J. Bolte, R. Glaser and S. Keppeler on:\n Semiclassical theory of spin-orbit interactions using spin coherent states: We reply to a Comment on our recently proposed semiclassical theory for\nsystems with spin-orbit interactions.", + "category": "nlin_CD" + }, + { + "text": "Quantum fluctuations stabilize an inverted pendulum: We explore analytically the quantum dynamics of a point mass pendulum using\nthe Heisenberg equation of motion. Choosing as variables the mean position of\nthe pendulum, a suitably defined generalised variance and a generalised\nskewness, we set up a dynamical system which reproduces the correct limits of\nsimple harmonic oscillator like and free rotor like behaviour. We then find the\nunexpected result that the quantum pendulum released from and near the inverted\nposition executes oscillatory motion around the classically unstable position\nprovided the initial wave packet has a variance much greater than the variance\nof the well known coherent state of the simple harmonic oscillator. The\nbehaviour of the dynamical system for the quantum pendulum is a higher\ndimensional analogue of the behaviour of the Kapitza pendulum where the point\nof support is vibrated vertically with a frequency higher than the critical\nvalue needed to stabilize the inverted position. A somewhat similar phenomenon\nhas recently been observed in the non equilibrium dynamics of a spin - 1\nBose-Einstein Condensate.", + "category": "nlin_CD" + }, + { + "text": "A Universal Map for Fractal Structures in Weak Solitary Wave\n Interactions: Fractal scatterings in weak solitary wave interactions is analyzed for\ngeneralized nonlinear Schr\\\"odiger equations (GNLS). Using asymptotic methods,\nthese weak interactions are reduced to a universal second-order map. This map\ngives the same fractal scattering patterns as those in the GNLS equations both\nqualitatively and quantitatively. Scaling laws of these fractals are also\nderived.", + "category": "nlin_CD" + }, + { + "text": "Desynchronization of systems of coupled Hindmarsh-Rose oscillators: It is widely assumed that neural activity related to synchronous rhythms of\nlarge portions of neurons in specific locations of the brain is responsible for\nthe pathology manifested in patients' uncontrolled tremor and other similar\ndiseases. To model such systems Hindmarsh-Rose (HR) oscillators are considered\nas appropriate as they mimic the qualitative behaviour of neuronal firing. Here\nwe consider a large number of identical HR-oscillators interacting through the\nmean field created by the corresponding components of all oscillators.\nIntroducing additional coupling by feedback of Pyragas type, proportional to\nthe difference between the current value of the mean-field and its value some\ntime in the past, Rosenblum and Pikovsky (Phys. Rev. E 70, 041904, 2004)\ndemonstrated that the desirable desynchronization could be achieved with\nappropriate set of parameters for the system. Following our experience with\nstabilization of unstable steady states in dynamical systems, we show that by\nintroducing a variable delay, desynchronization is obtainable for much wider\nrange of parameters and that at the same time it becomes more pronounced.", + "category": "nlin_CD" + }, + { + "text": "On the inadequacy of the logistic map for cryptographic applications: This paper analyzes the use of the logistic map for cryptographic\napplications. The most important characteristics of the logistic map are shown\nin order to prove the inconvenience of considering this map in the design of\nnew chaotic cryptosystems.", + "category": "nlin_CD" + }, + { + "text": "Collection of Master-Slave Synchronized Chaotic Systems: In this work the open-plus-closed-loop (OPCL) method of synchronization is\nused in order to synchronize the systems from the Sprott's collection of the\nsimplest chaotic systems. The method is general and we were looking for the\nsimplest coupling between master and slave system. The interval of parameters\nwere synchronization is achieved are obtained analytically using Routh-Hurwitz\nconditions. Detailed calculations and numerical simulation are given for the\nsystem I from the Sprott's collection. Working in the same manner for\nnon-linear systems based on ordinary differential equations the method can be\nadopted for the teaching of the topic.", + "category": "nlin_CD" + }, + { + "text": "Drastic facilitation of the onset of global chaos in a periodically\n driven Hamiltonian system due to an extremum in the dependence of\n eigenfrequency on energy: The Chirikov resonance-overlap criterion predicts the onset of global chaos\nif nonlinear resonances overlap in energy, which is conventionally assumed to\nrequire a non-small magnitude of perturbation. We show that, for a\ntime-periodic perturbation, the onset of global chaos may occur at unusually\n{\\it small} magnitudes of perturbation if the unperturbed system possesses more\nthan one separatrix. The relevant scenario is the combination of the overlap in\nthe phase space between resonances of the same order and their overlap in\nenergy with chaotic layers associated with separatrices of the unperturbed\nsystem. One of the most important manifestations of this effect is a drastic\nincrease of the energy range involved into the unbounded chaotic transport in\nspatially periodic system driven by a rather {\\it weak} time-periodic force,\nprovided the driving frequency approaches the extremal eigenfrequency or its\nharmonics. We develop the asymptotic theory and verify it in simulations.", + "category": "nlin_CD" + }, + { + "text": "On the detuned 2:4 resonance: We consider families of Hamiltonian systems in two degrees of freedom with an\nequilibrium in 1:2 resonance. Under detuning, this \"Fermi resonance\" typically\nleads to normal modes losing their stability through period-doubling\nbifurcations. For cubic potentials this concerns the short axial orbits and in\ngalactic dynamics the resulting stable periodic orbits are called \"banana\"\norbits. Galactic potentials are symmetric with respect to the co-ordinate\nplanes whence the potential -- and the normal form -- both have no cubic terms.\nThis $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$-symmetry turns the 1:2 resonance into a\nhigher order resonance and one therefore also speaks of the 2:4 resonance. In\nthis paper we study the 2:4 resonance in its own right, not restricted to\nnatural Hamiltonian systems where $H = T + V$ would consist of kinetic and\n(positional) potential energy. The short axial orbit then turns out to be\ndynamically stable everywhere except at a simultaneous bifurcation of banana\nand \"anti-banana\" orbits, while it is now the long axial orbit that loses and\nregains stability through two successive period-doubling bifurcations.", + "category": "nlin_CD" + }, + { + "text": "Anomalous correlators, \"ghost\" waves and nonlinear standing waves in the\n $\u03b2$-FPUT system: We show that Hamiltonian nonlinear dispersive wave systems with cubic\nnonlinearity and random initial data develop, during their evolution, anomalous\ncorrelators. These are responsible for the appearance of \"ghost\" excitations,\ni.e. those characterized by negative frequencies, in addition to the positive\nones predicted by the linear dispersion relation. We use generalization of the\nWick's decomposition and the wave turbulence theory to explain theoretically\nthe existence of anomalous correlators. We test our theory on the celebrated\n$\\beta$-Fermi-Pasta-Ulam-Tsingou chain and show that numerically measured\nvalues of the anomalous correlators agree, in the weakly nonlinear regime, with\nour analytical predictions. We also predict that similar phenomena will occur\nin other nonlinear systems dominated by nonlinear interactions, including\nsurface gravity waves. Our results pave the road to study phase correlations in\nthe Fourier space for weakly nonlinear dispersive wave systems.", + "category": "nlin_CD" + }, + { + "text": "A new model of variable-length coupled pendulums: from hyperchaos to\n superintegrability: This paper studies the dynamics and integrability of a variable-length\ncoupled pendulum system. The complexity of the model is presented by joining\nvarious numerical methods, such as the Poincar\\'e cross-sections,\nphase-parametric diagrams, and Lyapunov exponents spectra. We show that the\npresented model is hyperchaotic, which ensures its nonintegrability. We gave\nanalytical proof of this fact analyzing properties of the differential Galois\ngroup of variational equations along certain particular solutions of the\nsystem. We employ the Kovacic algorithm and its extension to dimension four to\nanalyze the differential Galois group. Amazingly enough, in the absence of the\ngravitational potential and for certain values of the parameters, the system\ncan exhibit chaotic, integrable, as well as superintegrable dynamics. To the\nbest of our knowledge, this is the first attempt to use the method of Lyapunov\nexponents in the systematic search for the first integrals of the system. We\nshow how to effectively apply the Lyapunov exponents as an indicator of\nintegrable dynamics. The explicit forms of integrable and superintegrable\nsystems are given.\n The article has been published in Nonlinear Dynamics, and the final version\nis available at this link: https://doi.org/10.1007/s11071-023-09253-5", + "category": "nlin_CD" + }, + { + "text": "Cycle expansions for intermittent maps: In a generic dynamical system chaos and regular motion coexist side by side,\nin different parts of the phase space. The border between these, where\ntrajectories are neither unstable nor stable but of marginal stability,\nmanifests itself through intermittency, dynamics where long periods of nearly\nregular motions are interrupted by irregular chaotic bursts. We discuss the\nPerron-Frobenius operator formalism for such systems, and show by means of a\n1-dimensional intermittent map that intermittency induces branch cuts in\ndynamical zeta functions. Marginality leads to long-time dynamical\ncorrelations, in contrast to the exponentially fast decorrelations of purely\nchaotic dynamics. We apply the periodic orbit theory to quantitative\ncharacterization of the associated power-law decays.", + "category": "nlin_CD" + }, + { + "text": "Recovery time after localized perturbations in complex dynamical\n networks: Maintaining the synchronous motion of dynamical systems interacting on\ncomplex networks is often critical to their functionality. However, real-world\nnetworked dynamical systems operating synchronously are prone to random\nperturbations driving the system to arbitrary states within the corresponding\nbasin of attraction, thereby leading to epochs of desynchronized dynamics with\na priori unknown durations. Thus, it is highly relevant to have an estimate of\nthe duration of such transient phases before the system returns to synchrony,\nfollowing a random perturbation to the dynamical state of any particular node\nof the network. We address this issue here by proposing the framework of\n\\emph{single-node recovery time} (SNRT) which provides an estimate of the\nrelative time scales underlying the transient dynamics of the nodes of a\nnetwork during its restoration to synchrony. We utilize this in differentiating\nthe particularly \\emph{slow} nodes of the network from the relatively\n\\emph{fast} nodes, thus identifying the critical nodes which when perturbed\nlead to significantly enlarged recovery time of the system before resuming\nsynchronized operation. Further, we reveal explicit relationships between the\nSNRT values of a network, and its \\emph{global relaxation time} when starting\nall the nodes from random initial conditions. We employ the proposed concept\nfor deducing microscopic relationships between topological features of nodes\nand their respective SNRT values. The framework of SNRT is further extended to\na measure of resilience of the different nodes of a networked dynamical system.\nWe demonstrate the potential of SNRT in networks of R\\\"{o}ssler oscillators on\nparadigmatic topologies and a model of the power grid of the United Kingdom\nwith second-order Kuramoto-type nodal dynamics illustrating the conceivable\npractical applicability of the proposed concept.", + "category": "nlin_CD" + }, + { + "text": "On asymptotic properties of some complex Lorenz-like systems: The classical Lorenz lowest order system of three nonlinear ordinary\ndifferential equations, capable of producing chaotic solutions, has been\ngeneralized by various authors in two main directions: (i) for number of\nequations larger than three (Curry1978) and (ii) for the case of complex\nvariables and parameters. Problems of laser physics and geophysical fluid\ndynamics (baroclinic instability, geodynamic theory, etc. - see the references)\ncan be related to this second aspect of generalization. In this paper we study\nthe asymptotic properties of some complex Lorenz systems, keeping in the mind\nthe physical basis of the model mathematical equations.", + "category": "nlin_CD" + }, + { + "text": "Sensitivity Analysis of Separation Time Along Weak Stability Boundary\n Transfers: This study analyzes the sensitivity of the dynamics around Weak Stability\nBoundary Transfers (WSBT) in the elliptical restricted three-body problem. With\nWSBTs increasing popularity for cislunar transfers, understanding its\ninherently chaotic dynamics becomes pivotal for guiding and navigating\ncooperative spacecrafts as well as detecting non-cooperative objects. We\nintroduce the notion of separation time to gauge the deviation of a point near\na nominal WSBT from the trajectory's vicinity. Employing the Cauchy-Green\ntensor to identify stretching directions in position and velocity, the\nseparation time, along with the Finite-Time Lyapunov Exponent are studied\nwithin a ball of state uncertainty scaled to typical orbit determination\nperformances.", + "category": "nlin_CD" + }, + { + "text": "Turbulent boundary layer equations: We study a boundary layer problem for the Navier-Stokes-alpha model obtaining\na generalization of the Prandtl equations conjectured to represent the averaged\nflow in a turbulent boundary layer. We solve the equations for the\nsemi-infinite plate, both theoretically and numerically. The latter solutions\nagree with some experimental data in the turbulent boundary layer.", + "category": "nlin_CD" + }, + { + "text": "On universality of algebraic decays in Hamiltonian systems: Hamiltonian systems with a mixed phase space typically exhibit an algebraic\ndecay of correlations and of Poincare' recurrences, with numerical experiments\nover finite times showing system-dependent power-law exponents. We conjecture\nthe existence of a universal asymptotic decay based on results for a Markov\ntree model with random scaling factors for the transition probabilities.\nNumerical simulations for different Hamiltonian systems support this conjecture\nand permit the determination of the universal exponent.", + "category": "nlin_CD" + }, + { + "text": "Amplitude death in a ring of nonidentical nonlinear oscillators with\n unidirectional coupling: We study the collective behaviors in a ring of coupled nonidentical nonlinear\noscillators with unidirectional coupling, of which natural frequencies are\ndistributed in a random way. We find the amplitude death phenomena in the case\nof unidirectional couplings and discuss the differences between the cases of\nbidirectional and unidirectional couplings. There are three main differences;\nthere exists neither partial amplitude death nor local clustering behavior but\noblique line structure which represents directional signal flow on the\nspatio-temporal patterns in the unidirectional coupling case. The\nunidirectional coupling has the advantage of easily obtaining global amplitude\ndeath in a ring of coupled oscillators with randomly distributed natural\nfrequency. Finally, we explain the results using the eigenvalue analysis of\nJacobian matrix at the origin and also discuss the transition of dynamical\nbehavior coming from connection structure as coupling strength increases.", + "category": "nlin_CD" + }, + { + "text": "Cryptanalysis of a chaotic block cipher with external key and its\n improved version: Recently, Pareek et al. proposed a symmetric key block cipher using multiple\none-dimensional chaotic maps. This paper reports some new findings on the\nsecurity problems of this kind of chaotic cipher: 1) a number of weak keys\nexists; 2) some important intermediate data of the cipher are not sufficiently\nrandom; 3) the whole secret key can be broken by a known-plaintext attack with\nonly 120 consecutive known plain-bytes in one known plaintext. In addition, it\nis pointed out that an improved version of the chaotic cipher proposed by Wei\net al. still suffers from all the same security defects.", + "category": "nlin_CD" + }, + { + "text": "Lagrangian transport through surfaces in volume-preserving flows: Advective transport of scalar quantities through surfaces is of fundamental\nimportance in many scientific applications. From the Eulerian perspective of\nthe surface it can be quantified by the well-known integral of the flux\ndensity. The recent development of highly accurate semi-Lagrangian methods for\nsolving scalar conservation laws and of Lagrangian approaches to coherent\nstructures in turbulent (geophysical) fluid flows necessitate a new approach to\ntransport from the (Lagrangian) material perspective. We present a Lagrangian\nframework for calculating transport of conserved quantities through a given\nsurface in $n$-dimensional, fully aperiodic, volume-preserving flows. Our\napproach does not involve any dynamical assumptions on the surface or its\nboundary.", + "category": "nlin_CD" + }, + { + "text": "Big Entropy Fluctuations in Nonequilibrium Steady State: A Simple Model\n with Gauss Heat Bath: Large entropy fluctuations in a nonequilibrium steady state of classical\nmechanics were studied in extensive numerical experiments on a simple 2-freedom\nmodel with the so-called Gauss time-reversible thermostat. The local\nfluctuations (on a set of fixed trajectory segments) from the average heat\nentropy absorbed in thermostat were found to be non-Gaussian. Approximately,\nthe fluctuations can be discribed by a two-Gaussian distribution with a\ncrossover independent of the segment length and the number of trajectories\n('particles'). The distribution itself does depend on both, approaching the\nsingle standard Gaussian distribution as any of those parameters increases. The\nglobal time-dependent fluctuations turned out to be qualitatively different in\nthat they have a strict upper bound much less than the average entropy\nproduction. Thus, unlike the equilibrium steady state, the recovery of the\ninitial low entropy becomes impossible, after a sufficiently long time, even in\nthe largest fluctuations. However, preliminary numerical experiments and the\ntheoretical estimates in the special case of the critical dynamics with\nsuperdiffusion suggest the existence of infinitely many Poincar\\'e recurrences\nto the initial state and beyond. This is a new interesting phenomenon to be\nfarther studied together with some other open questions. Relation of this\nparticular example of nonequilibrium steady state to a long-standing persistent\ncontroversy over statistical 'irreversibility', or the notorious 'time arrow',\nis also discussed. In conclusion, an unsolved problem of the origin of the\ncausality 'principle' is touched upon.", + "category": "nlin_CD" + }, + { + "text": "Detecting Generalized Synchronization Between Chaotic Signals: A\n Kernel-based Approach: A unified framework for analyzing generalized synchronization in coupled\nchaotic systems from data is proposed. The key of the proposed approach is the\nuse of the kernel methods recently developed in the field of machine learning.\nSeveral successful applications are presented, which show the capability of the\nkernel-based approach for detecting generalized synchronization. It is also\nshown that the dynamical change of the coupling coefficient between two chaotic\nsystems can be captured by the proposed approach.", + "category": "nlin_CD" + }, + { + "text": "Entropy production and Lyapunov instability at the onset of turbulent\n convection: Computer simulations of a compressible fluid, convecting heat in two\ndimensions, suggest that, within a range of Rayleigh numbers, two distinctly\ndifferent, but stable, time-dependent flow morphologies are possible. The\nsimpler of the flows has two characteristic frequencies: the rotation frequency\nof the convecting rolls, and the vertical oscillation frequency of the rolls.\nObservables, such as the heat flux, have a simple-periodic (harmonic) time\ndependence. The more complex flow has at least one additional characteristic\nfrequency -- the horizontal frequency of the cold, downward- and the warm,\nupward-flowing plumes. Observables of this latter flow have a broadband\nfrequency distribution. The two flow morphologies, at the same Rayleigh number,\nhave different rates of entropy production and different Lyapunov exponents.\nThe simpler \"harmonic\" flow transports more heat (produces entropy at a greater\nrate), whereas the more complex \"chaotic\" flow has a larger maximum Lyapunov\nexponent (corresponding to a larger rate of phase-space information loss). A\nlinear combination of these two rates is invariant for the two flow\nmorphologies over the entire range of Rayleigh numbers for which the flows\ncoexist, suggesting a relation between the two rates near the onset of\nconvective turbulence.", + "category": "nlin_CD" + }, + { + "text": "Timing of Transients: Quantifying Reaching Times and Transient Behavior\n in Complex Systems: When quantifying the time spent in the transient of a complex dynamical\nsystem, the fundamental problem is that for a large class of systems the actual\ntime for reaching an attractor is infinite. Common methods for dealing with\nthis problem usually introduce three additional problems: non-invariance,\nphysical interpretation, and discontinuities, calling for carefully designed\nmethods for quantifying transients.\n In this article, we discuss how the aforementioned problems emerge and\npropose two novel metrics, Regularized Reaching Time ($T_{RR}$) and Area under\nDistance Curve (AUDIC), to solve them, capturing two complementary aspects of\nthe transient dynamics.\n $T_{RR}$ quantifies the additional time (positive or negative) that a\ntrajectory starting at a chosen initial condition needs to reach the attractor\nafter a reference trajectory has already arrived there. A positive or negative\nvalue means that it arrives by this much earlier or later than the reference.\nBecause $T_{RR}$ is an analysis of return times after shocks, it is a\nsystematic approach to the concept of critical slowing down [1]; hence it is\nnaturally an early-warning signal [2] for bifurcations when central statistics\nover distributions of initial conditions are used.\n AUDIC is the distance of the trajectory to the attractor integrated over\ntime. Complementary to $T_{RR}$, it measures which trajectories are reluctant,\ni.e. stay away from the attractor for long, or eager to approach it right away.\n (... shortened for arxiv listing, full abstract in paper ...) New features in\nthese models can be uncovered, including the surprising regularity of the\nRoessler system's basin of attraction even in the regime of a chaotic\nattractor. Additionally, we demonstrate the critical slowing down\ninterpretation by presenting the metrics' sensitivity to prebifurcational\nchange and thus how they act as early-warning signals.", + "category": "nlin_CD" + }, + { + "text": "Intensity distribution of non-linear scattering states: We investigate the interplay between coherent effects characteristic of the\npropagation of linear waves, the non-linear effects due to interactions, and\nthe quantum manifestations of classical chaos due to geometrical confinement,\nas they arise in the context of the transport of Bose-Einstein condensates. We\nspecifically show that, extending standard methods for non-interacting systems,\nthe body of the statistical distribution of intensities for scattering states\nsolving the Gross-Pitaevskii equation is very well described by a local\nGaussian ansatz with a position-dependent variance. We propose a semiclassical\napproach based on interfering classical paths to fix the single parameter\ndescribing the universal deviations from a global Gaussian distribution. Being\ntail effects, rare events like rogue waves characteristic of non-linear field\nequations do not affect our results.", + "category": "nlin_CD" + }, + { + "text": "A Plethora of Strange Nonchaotic Attractors: We show that it is possible to devise a large class of skew--product\ndynamical systems which have strange nonchaotic attractors (SNAs): the dynamics\nis asymptotically on fractal attractors and the largest Lyapunov exponent is\nnonpositive. Furthermore, we show that quasiperiodic forcing, which has been a\nhallmark of essentially allhitherto known examples of such dynamics is {\\it\nnot} necessary for the creation of SNAs.", + "category": "nlin_CD" + }, + { + "text": "Stochastic mean field formulation of the dynamics of diluted neural\n networks: We consider pulse-coupled Leaky Integrate-and-Fire neural networks with\nrandomly distributed synaptic couplings. This random dilution induces\nfluctuations in the evolution of the macroscopic variables and deterministic\nchaos at the microscopic level. Our main aim is to mimic the effect of the\ndilution as a noise source acting on the dynamics of a globally coupled\nnon-chaotic system. Indeed, the evolution of a diluted neural network can be\nwell approximated as a fully pulse coupled network, where each neuron is driven\nby a mean synaptic current plus additive noise. These terms represent the\naverage and the fluctuations of the synaptic currents acting on the single\nneurons in the diluted system. The main microscopic and macroscopic dynamical\nfeatures can be retrieved with this stochastic approximation. Furthermore, the\nmicroscopic stability of the diluted network can be also reproduced, as\ndemonstrated from the almost coincidence of the measured Lyapunov exponents in\nthe deterministic and stochastic cases for an ample range of system sizes. Our\nresults strongly suggest that the fluctuations in the synaptic currents are\nresponsible for the emergence of chaos in this class of pulse coupled networks.", + "category": "nlin_CD" + }, + { + "text": "Unraveling the Chaos-land and its organization in the Rabinovich System: A suite of analytical and computational techniques based on symbolic\nrepresentations of simple and complex dynamics, is further developed and\nemployed to unravel the global organization of bi-parametric structures that\nunderlie the emergence of chaos in a simplified resonantly coupled wave triplet\nsystem, known as the Rabinovich system. Bi-parametric scans reveal the stunning\nintricacy and intramural connections between homoclinic and heteroclinic\nconnections, and codimension-2 Bykov T-points and saddle structures, which are\nthe prime organizing centers of complexity of the bifurcation unfolding of the\ngiven system. This suite includes Deterministic Chaos Prospector (DCP) to sweep\nand effectively identify regions of simple (Morse-Smale) and chaotic\nstructurally unstable dynamics in the system. Our analysis provides striking\nnew insights into the complex behaviors exhibited by this and similar systems.", + "category": "nlin_CD" + }, + { + "text": "A Non-Equilibrium Defect-Unbinding Transition: Defect Trajectories and\n Loop Statistics: In a Ginzburg-Landau model for parametrically driven waves a transition\nbetween a state of ordered and one of disordered spatio-temporal defect chaos\nis found. To characterize the two different chaotic states and to get insight\ninto the break-down of the order, the trajectories of the defects are tracked\nin detail. Since the defects are always created and annihilated in pairs the\ntrajectories form loops in space time. The probability distribution functions\nfor the size of the loops and the number of defects involved in them undergo a\ntransition from exponential decay in the ordered regime to a power-law decay in\nthe disordered regime. These power laws are also found in a simple lattice\nmodel of randomly created defect pairs that diffuse and annihilate upon\ncollision.", + "category": "nlin_CD" + }, + { + "text": "Designing two-dimensional limit-cycle oscillators with prescribed\n trajectories and phase-response characteristics: We propose a method for designing two-dimensional limit-cycle oscillators\nwith prescribed periodic trajectories and phase response properties based on\nthe phase reduction theory, which gives a concise description of\nweakly-perturbed limit-cycle oscillators and is widely used in the analysis of\nsynchronization dynamics. We develop an algorithm for designing the vector\nfield with a stable limit cycle, which possesses a given shape and also a given\nphase sensitivity function. The vector field of the limit-cycle oscillator is\napproximated by polynomials whose coefficients are estimated by convex\noptimization. Linear stability of the limit cycle is ensured by introducing an\nupper bound to the Floquet exponent. The validity of the proposed method is\nverified numerically by designing several types of two-dimensional existing and\nartificial oscillators. As applications, we first design a limit-cycle\noscillator with an artificial star-shaped periodic trajectory and demonstrate\nglobal entrainment. We then design a limit-cycle oscillator with an artificial\nhigh-harmonic phase sensitivity function and demonstrate multistable\nentrainment caused by a high-frequency periodic input.", + "category": "nlin_CD" + }, + { + "text": "Statistical and dynamical properties of the quantum triangle map: We study the statistical and dynamical properties of the quantum triangle\nmap, whose classical counterpart can exhibit ergodic and mixing dynamics, but\nis never chaotic. Numerical results show that ergodicity is a sufficient\ncondition for spectrum and eigenfunctions to follow the prediction of Random\nMatrix Theory, even though the underlying classical dynamics is not chaotic. On\nthe other hand, dynamical quantities such as the out-of-time-ordered correlator\n(OTOC) and the number of harmonics, exhibit a growth rate vanishing in the\nsemiclassical limit, in agreement with the fact that classical dynamics has\nzero Lyapunov exponent. Our finding show that, while spectral statistics can be\nused to detect ergodicity, OTOC and number of harmonics are diagnostics of\nchaos.", + "category": "nlin_CD" + }, + { + "text": "Scaling regimes of 2d turbulence with power law stirring: theories\n versus numerical experiments: We inquire the statistical properties of the pair formed by the Navier-Stokes\nequation for an incompressible velocity field and the advection-diffusion\nequation for a scalar field transported in the same flow in two dimensions\n(2d). The system is in a regime of fully developed turbulence stirred by\nforcing fields with Gaussian statistics, white-noise in time and self-similar\nin space. In this setting and if the stirring is concentrated at small spatial\nscales as if due to thermal fluctuations, it is possible to carry out a\nfirst-principle ultra-violet renormalization group analysis of the scaling\nbehavior of the model.\n Kraichnan's phenomenological theory of two dimensional turbulence upholds the\nexistence of an inertial range characterized by inverse energy transfer at\nscales larger than the stirring one. For our model Kraichnan's theory, however,\nimplies scaling predictions radically discordant from the renormalization group\nresults. We perform accurate numerical experiments to assess the actual\nstatistical properties of 2d-turbulence with power-law stirring. Our results\nclearly indicate that an adapted version of Kraichnan's theory is consistent\nwith the observed phenomenology. We also provide some theoretical scenarios to\naccount for the discrepancy between renormalization group analysis and the\nobserved phenomenology.", + "category": "nlin_CD" + }, + { + "text": "Long-time signatures of short-time dynamics in decaying quantum-chaotic\n systems: We analyze the decay of classically chaotic quantum systems in the presence\nof fast ballistic escape routes on the Ehrenfest time scale. For a continuous\nexcitation process, the form factor of the decay cross section deviates from\nthe universal random-matrix result on the Heisenberg time scale, i.e. for times\nmuch larger than the time for ballistic escape. We derive an exact analytical\ndescription and compare our results with numerical simulations for a dynamical\nmodel.", + "category": "nlin_CD" + }, + { + "text": "A Phase-Space Approach for Propagating Field-Field Correlation Functions: We show that radiation from complex and inherently random but correlated wave\nsources can be modelled efficiently by using an approach based on the Wigner\ndistribution function. Our method exploits the connection between correlation\nfunctions and theWigner function and admits in its simplest approximation a\ndirect representation in terms of the evolution of ray densities in phase\nspace. We show that next leading order corrections to the ray-tracing\napproximation lead to Airy-function type phase space propagators. By exploiting\nthe exact Wigner function propagator, inherently wave-like effects such as\nevanescent decay or radiation from more heterogeneous sources as well as\ndiffraction and reflections can be included and analysed. We discuss in\nparticular the role of evanescent waves in the near-field of non-paraxial\nsources and give explicit expressions for the growth rate of the correlation\nlength as function of the distance from the source. Furthermore, results for\nthe reflection of partially coherent sources from flat mirrors are given. We\nfocus here on electromagnetic sources at microwave frequencies and modelling\nefforts in the context of electromagnetic compatibility.", + "category": "nlin_CD" + }, + { + "text": "The dynamical temperature and the standard map: Numerical experiments with the standard map at high values of the\nstochasticity parameter reveal the existence of simple analytical relations\nconnecting the volume and the dynamical temperature of the chaotic component of\nthe phase space.", + "category": "nlin_CD" + }, + { + "text": "Synchronized bursts following instability of synchronous spiking in\n chaotic neuronal networks: We report on the origin of synchronized bursting dynamics in various networks\nof neural spiking oscillators, when a certain threshold in coupling strength is\nexceeded. These ensembles synchronize at relatively low coupling strength and\nlose synchronization at stronger coupling via spatio-temporal intermittency.\nThe latter transition triggers multiple-timescale dynamics, which results in\nsynchronized bursting with a fractal-like spatio-temporal pattern of spiking.\nImplementation of an appropriate technique of separating oscillations on\ndifferent time-scales allows for quantitative analysis of this phenomenon. We\nshow, that this phenomenon is generic for various network topologies from\nregular to small-world and scale-free ones and for different types of coupling.", + "category": "nlin_CD" + }, + { + "text": "Excitable systems with noise and delay with applications to control:\n renewal theory approach: We present an approach for the analytical treatment of excitable systems with\nnoise-induced dynamics in the presence of time delay. An excitable system is\nmodeled as a bistable system with a time delay, while another delay enters as a\ncontrol term taken after [Pyragas 1992] as a difference between the current\nsystem state and its state \"tau\" time units before. This approach combines the\nelements of renewal theory to estimate the essential features of the resulting\nstochastic process as functions of the parameters of the controlling term.", + "category": "nlin_CD" + }, + { + "text": "Noise-enhanced trapping in chaotic scattering: We show that noise enhances the trapping of trajectories in scattering\nsystems. In fully chaotic systems, the decay rate can decrease with increasing\nnoise due to a generic mismatch between the noiseless escape rate and the value\npredicted by the Liouville measure of the exit set. In Hamiltonian systems with\nmixed phase space we show that noise leads to a slower algebraic decay due to\ntrajectories performing a random walk inside Kolmogorov-Arnold-Moser islands.\nWe argue that these noise-enhanced trapping mechanisms exist in most scattering\nsystems and are likely to be dominant for small noise intensities, which is\nconfirmed through a detailed investigation in the Henon map. Our results can be\ntested in fluid experiments, affect the fractal Weyl's law of quantum systems,\nand modify the estimations of chemical reaction rates based on phase-space\ntransition state theory.", + "category": "nlin_CD" + }, + { + "text": "The paradox of infinitesimal granularity: Chaos and the reversibility of\n time in Newton's theory of gravity: The fundamental laws of physics are time-symmetric, but our macroscopic\nexperience contradicts this. The time reversibility paradox is partly a\nconsequence of the unpredictability of Newton's equations of motion. We measure\nthe dependence of the fraction of irreversible, gravitational N-body systems on\nnumerical precision and find that it scales as a power law. The stochastic wave\npacket reduction postulate then introduces fundamental uncertainties in the\nCartesian phase space coordinates that propagate through classical three-body\ndynamics to macroscopic scales within the triple's lifetime. The spontaneous\ncollapse of the wave function then drives the global chaotic behavior of the\nUniverse through the superposition of triple systems (and probably multi-body\nsystems). The paradox of infinitesimal granularity then arises from the\nsuperposition principle, which states that any multi-body system is composed of\nan ensemble of three-body problems.", + "category": "nlin_CD" + }, + { + "text": "Functional renormalization-group approach to decaying turbulence: We reconsider the functional renormalization-group (FRG) approach to decaying\nBurgers turbulence, and extend it to decaying Navier-Stokes and\nSurface-Quasi-Geostrophic turbulence. The method is based on a renormalized\nsmall-time expansion, equivalent to a loop expansion, and naturally produces a\ndissipative anomaly and a cascade after a finite time. We explicitly calculate\nand analyze the one-loop FRG equations in the zero-viscosity limit as a\nfunction of the dimension. For Burgers they reproduce the FRG equation obtained\nin the context of random manifolds, extending previous results of one of us.\nBreakdown of energy conservation due to shocks and the appearance of a direct\nenergy cascade corresponds to failure of dimensional reduction in the context\nof disordered systems. For Navier-Stokes in three dimensions, the\nvelocity-velocity correlation function acquires a linear dependence on the\ndistance, zeta_2=1, in the inertial range, instead of Kolmogorov's zeta_2=2/3;\nhowever the possibility remains for corrections at two- or higher-loop order.\nIn two dimensions, we obtain a numerical solution which conserves energy and\nexhibits an inverse cascade, with explicit analytical results both for large\nand small distances, in agreement with the scaling proposed by Batchelor. In\nlarge dimensions, the one-loop FRG equation for Navier-Stokes converges to that\nof Burgers.", + "category": "nlin_CD" + }, + { + "text": "Stabilisation of long-period periodic orbits using time-delayed feedback\n control: The Pyragas method of feedback control has attracted much interest as a\nmethod of stabilising unstable periodic orbits in a number of situations. We\nshow that a time-delayed feedback control similar to the Pyragas method can be\nused to stabilise periodic orbits with arbitrarily large period, specifically\nthose resulting from a resonant bifurcation of a heteroclinic cycle. Our\nanalysis reduces the infinite-dimensional delay-equation governing the system\nwith feedback to a three-dimensional map, by making certain assumptions about\nthe form of the solutions. The stability of a fixed point in this map\ncorresponds to the stability of the periodic orbit in the flow, and can be\ncomputed analytically. We compare the analytic results to a numerical example\nand find very good agreement.", + "category": "nlin_CD" + }, + { + "text": "A Mechanical Analog of the Two-bounce Resonance of Solitary Waves:\n Modeling and Experiment: We describe a simple mechanical system, a ball rolling along a\nspecially-designed landscape, that mimics the dynamics of a well known\nphenomenon, the two-bounce resonance of solitary wave collisions, that has been\nseen in countless numerical simulations but never in the laboratory. We provide\na brief history of the solitary wave problem, stressing the fundamental role\ncollective-coordinate models played in understanding this phenomenon. We derive\nthe equations governing the motion of a point particle confined to such a\nsurface and then design a surface on which to roll the ball, such that its\nmotion will evolve under the same equations that approximately govern solitary\nwave collisions. We report on physical experiments, carried out in an\nundergraduate applied mathematics course, that seem to verify one aspect of\nchaotic scattering, the so-called two-bounce resonance.", + "category": "nlin_CD" + }, + { + "text": "Simple models of bouncing ball dynamics and their comparison: Nonlinear dynamics of a bouncing ball moving in gravitational field and\ncolliding with a moving limiter is considered. Several simple models of table\nmotion are studied and compared. Dependence of displacement of the table on\ntime, approximating sinusoidal motion and making analytical computations\npossible, is assumed as quadratic and cubic functions of time, respectively.", + "category": "nlin_CD" + }, + { + "text": "Intermittency effects in Burgers equation driven by thermal noise: For the Burgers equation driven by thermal noise leading asymptotics of pair\nand high-order correlators of the velocity field are found for finite times and\nlarge distances. It is shown that the intermittency takes place: some\ncorrelators are much larger than their reducible parts.", + "category": "nlin_CD" + }, + { + "text": "Morphological Image Analysis of Quantum Motion in Billiards: Morphological image analysis is applied to the time evolution of the\nprobability distribution of a quantum particle moving in two and\nthree-dimensional billiards. It is shown that the time-averaged Euler\ncharacteristic of the probability density provides a well defined quantity to\ndistinguish between classically integrable and non-integrable billiards. In\nthree dimensions the time-averaged mean breadth of the probability density may\nalso be used for this purpose.", + "category": "nlin_CD" + }, + { + "text": "From quasiperiodicity to high-dimensional chaos without intermediate\n low-dimensional chaos: We study and characterize a direct route to high-dimensional chaos (i.e. not\nimplying an intermediate low-dimensional attractor) of a system composed out of\nthree coupled Lorenz oscillators. A geometric analysis of this\nmedium-dimensional dynamical system is carried out through a variety of\nnumerical quantitative and qualitative techniques, that ultimately lead to the\nreconstruction of the route. The main finding is that the transition is\norganized by a heteroclinic explosion. The observed scenario resembles the\nclassical route to chaos via homoclinic explosion of the Lorenz model.", + "category": "nlin_CD" + }, + { + "text": "On the Implementation of the 0-1 Test for Chaos: In this paper we address practical aspects of the implementation of the 0-1\ntest for chaos in deterministic systems. In addition, we present a new\nformulation of the test which significantly increases its sensitivity. The test\ncan be viewed as a method to distill a binary quantity from the power spectrum.\nThe implementation is guided by recent results from the theoretical\njustification of the test as well as by exploring better statistical methods to\ndetermine the binary quantities. We give several examples to illustrate the\nimprovement.", + "category": "nlin_CD" + }, + { + "text": "General mechanism for amplitude death in coupled systems: We introduce a general mechanism for amplitude death in coupled\nsynchronizable dynamical systems. It is known that when two systems are coupled\ndirectly, they can synchronize under suitable conditions. When an indirect\nfeedback coupling through an environment or an external system is introduced in\nthem, it is found to induce a tendency for anti-synchronization. We show that,\nfor sufficient strengths, these two competing effects can lead to amplitude\ndeath. We provide a general stability analysis that gives the threshold values\nfor onset of amplitude death. We study in detail the nature of the transition\nto death in several specific cases and find that the transitions can be of two\ntypes - continuous and discontinuous. By choosing a variety of dynamics for\nexample, periodic, chaotic, hyper chaotic, and time-delay systems, we\nillustrate that this mechanism is quite general and works for different types\nof direct coupling, such as diffusive, replacement, and synaptic couplings and\nfor different damped dynamics of the environment.", + "category": "nlin_CD" + }, + { + "text": "Fluctuation-response relation in turbulent systems: We address the problem of measuring time-properties of Response Functions\n(Green functions) in Gaussian models (Orszag-McLaughin) and strongly\nnon-Gaussian models (shell models for turbulence). We introduce the concept of\n{\\it halving time statistics} to have a statistically stable tool to quantify\nthe time decay of Response Functions and Generalized Response Functions of high\norder. We show numerically that in shell models for three dimensional\nturbulence Response Functions are inertial range quantities. This is a strong\nindication that the invariant measure describing the shell-velocity\nfluctuations is characterized by short range interactions between neighboring\nshells.", + "category": "nlin_CD" + }, + { + "text": "Non-smooth model and numerical analysis of a friction driven structure\n for piezoelectric motors: In this contribution, typical friction driven structures are summarized and\npresented considering the mechanical structures and operation principles of\ndifferent types of piezoelectric motors. A two degree-of-freedom dynamic model\nwith one unilateral frictional contact is built for one of the friction driven\nstructures. Different contact regimes and the transitions between them are\nidentified and analyzed. Numerical simulations are conducted to find out\ndifferent operation modes of the system concerning the sequence of contact\nregimes in one steady state period. The influences of parameters on the\noperation modes and corresponding steady state characteristics are also\nexplored. Some advice are then given in terms of the design of friction driven\nstructures and piezoelectric motors.", + "category": "nlin_CD" + }, + { + "text": "Chaos suppression in the parametrically driven Lorenz system: We predict theoretically and verify experimentally the suppression of chaos\nin the Lorenz system driven by a high-frequency periodic or stochastic\nparametric force. We derive the theoretical criteria for chaos suppression and\nverify that they are in a good agreement with the results of numerical\nsimulations and the experimental data obtained for an analog electronic\ncircuit.", + "category": "nlin_CD" + }, + { + "text": "Coexisting synchronous and asynchronous states in locally coupled array\n of oscillators by partial self-feedback control: We report the emergence of coexisting synchronous and asynchronous\nsubpopulations of oscillators in one dimensional arrays of identical\noscillators by applying a self-feedback control. When a self-feedback is\napplied to a subpopulation of the array, similar to chimera states, it splits\ninto two/more sub-subpopulations coexisting in coherent and incoherent states\nfor a range of self-feedback strength. By tuning the coupling between the\nnearest neighbors and the amount of self-feedback in the perturbed\nsubpopulation, the size of the coherent and the incoherent sub-subpopulations\nin the array can be controlled, although the exact size of them is\nunpredictable. We present numerical evidence using the Landau-Stuart (LS)\nsystem and the Kuramoto-Sakaguchi (KS) phase model.", + "category": "nlin_CD" + }, + { + "text": "Dynamic Phase Transition from Localized to Spatiotemporal Chaos in\n Coupled Circle Map with Feedback: We investigate coupled circle maps in presence of feedback and explore\nvarious dynamical phases observed in this system of coupled high dimensional\nmaps. We observe an interesting transition from localized chaos to\nspatiotemporal chaos. We study this transition as a dynamic phase transition.\nWe observe that persistence acts as an excellent quantifier to describe this\ntransition. Taking the location of the fixed point of circle map (which does\nnot change with feedback) as a reference point, we compute number of sites\nwhich have been greater than (less than) the fixed point till time t. Though\nlocal dynamics is high-dimensional in this case this definition of persistence\nwhich tracks a single variable is an excellent quantifier for this transition.\nIn most cases, we also obtain a well defined persistence exponent at the\ncritical point and observe conventional scaling as seen in second order phase\ntransitions. This indicates that persistence could work as good order parameter\nfor transitions from fully or partially arrested phase. We also give an\nexplanation of gaps in eigenvalue spectrum of the Jacobian of localized state.", + "category": "nlin_CD" + }, + { + "text": "On WKB Series for the Radial Kepler Problem: We obtain the rigorous WKB expansion to all orders for the radial Kepler\nproblem, using the residue calculus in evaluating the WKB quantization\ncondition in terms of a complex contour integral in the complexified coordinate\nplane. The procedure yields the exact energy spectrum of this Schr\\\"odinger\neigenvalue problem and thus resolves the controversies around the so-called\n\"Langer correction\". The problem is nontrivial also because there are only a\nfew systems for which all orders of the WKB series can be calculated, yielding\na convergent series whose sum is equal to the exact result, and thus sheds new\nlight to similar and more difficult problems.", + "category": "nlin_CD" + }, + { + "text": "The reflection-antisymmetric counterpart of the K\u00e1rm\u00e1n-Howarth\n dynamical equation: We study the isotropic, helical component in homogeneous turbulence using\nstatistical objects which have the correct symmetry and parity properties.\nUsing these objects we derive an analogue of the K\\'arm\\'an-Howarth equation,\nthat arises due to parity violation in isotropic flows. The main equation we\nobtain is consistent with the results of O. Chkhetiani [JETP, 63, 768, (1996)]\nand\n V.S. L'vov et al. [chao-dyn/9705016,\n (1997)] but is derived using only velocity correlations, with no direct\nconsideration of the vorticity or helicity. This alternative formulation offers\nan advantage to both experimental and numerical measurements. We also\npostulate, under the assumption of self-similarity, the existence of a\nhierarchy of scaling exponents for helical velocity correlation functions of\narbitrary order, analogous to the\n Kolmogorov 1941 prediction for the scaling exponents of velocity structure\nfunction.", + "category": "nlin_CD" + }, + { + "text": "Chirikov and Nekhoroshev diffusion estimates: bridging the two sides of\n the river: We present theoretical and numerical results pointing towards a strong\nconnection between the estimates for the diffusion rate along simple resonances\nin multidimensional nonlinear Hamiltonian systems that can be obtained using\nthe heuristic theory of Chirikov and a more formal one due to Nekhoroshev. We\nshow that, despite a wide-spread impression, the two theories are complementary\nrather than antagonist. Indeed, although Chirikov's 1979 review has thousands\nof citations, almost all of them refer to topics such as the resonance overlap\ncriterion, fast diffusion, the Standard or Whisker Map, and not to the\nconstructive theory providing a formula to measure diffusion along a single\nresonance. However, as will be demonstrated explicitly below, Chirikov's\nformula provides values of the diffusion coefficient which are quite well\ncomparable to the numerically computed ones, provided that it is implemented on\nthe so-called optimal normal form derived as in the analytic part of\nNekhoroshev's theorem. On the other hand, Chirikov's formula yields unrealistic\nvalues of the diffusion coefficient, in particular for very small values of the\nperturbation, when used in the original Hamiltonian instead of the optimal\nnormal form. In the present paper, we take advantage of this complementarity in\norder to obtain accurate theoretical predictions for the local value of the\ndiffusion coefficient along a resonance in a specific 3DoF nearly integrable\nHamiltonian system. Besides, we compute numerically the diffusion coefficient\nand a full comparison of all estimates is made for ten values of the\nperturbation parameter, showing a very satisfactory agreement.", + "category": "nlin_CD" + }, + { + "text": "A study of the double pendulum using polynomial optimization: In dynamical systems governed by differential equations, a guarantee that\ntrajectories emanating from a given set of initial conditions do not enter\nanother given set can be obtained by constructing a barrier function that\nsatisfies certain inequalities on phase space. Often these inequalities amount\nto nonnegativity of polynomials and can be enforced using sum-of-squares\nconditions, in which case barrier functions can be constructed computationally\nusing convex optimization over polynomials. To study how well such computations\ncan characterize sets of initial conditions in a chaotic system, we use the\nundamped double pendulum as an example and ask which stationary initial\npositions do not lead to flipping of the pendulum within a chosen time window.\nComputations give semialgebraic sets that are close inner approximations to the\nfractal set of all such initial positions.", + "category": "nlin_CD" + }, + { + "text": "Defining Chaos: In this paper we propose, discuss and illustrate a computationally feasible\ndefinition of chaos which can be applied very generally to situations that are\ncommonly encountered, including attractors, repellers and non-periodically\nforced systems. This definition is based on an entropy-like quantity, which we\ncall \"expansion entropy\", and we define chaos as occurring when this quantity\nis positive. We relate and compare expansion entropy to the well-known concept\nof topological entropy, to which it is equivalent under appropriate conditions.\nWe also present example illustrations, discuss computational implementations,\nand point out issues arising from attempts at giving definitions of chaos that\nare not entropy-based.", + "category": "nlin_CD" + }, + { + "text": "Alpha-modeling strategy for LES of turbulent mixing: The $\\alpha$-modeling strategy is followed to derive a new subgrid\nparameterization of the turbulent stress tensor in large-eddy simulation (LES).\nThe LES-$\\alpha$ modeling yields an explicitly filtered subgrid\nparameterization which contains the filtered nonlinear gradient model as well\nas a model which represents `Leray-regularization'. The LES-$\\alpha$ model is\ncompared with similarity and eddy-viscosity models that also use the dynamic\nprocedure. Numerical simulations of a turbulent mixing layer are performed\nusing both a second order, and a fourth order accurate finite volume\ndiscretization. The Leray model emerges as the most accurate, robust and\ncomputationally efficient among the three LES-$\\alpha$ subgrid\nparameterizations for the turbulent mixing layer. The evolution of the resolved\nkinetic energy is analyzed and the various subgrid-model contributions to it\nare identified. By comparing LES-$\\alpha$ at different subgrid resolutions, an\nimpression of finite volume discretization error dynamics is obtained.", + "category": "nlin_CD" + }, + { + "text": "Dynamical complexity as a proxy for the network degree distribution: We explore the relation between the topological relevance of a node in a\ncomplex network and the individual dynamics it exhibits. When the system is\nweakly coupled, the effect of the coupling strength against the dynamical\ncomplexity of the nodes is found to be a function of their topological role,\nwith nodes of higher degree displaying lower levels of complexity. We provide\nseveral examples of theoretical models of chaotic oscillators, pulse-coupled\nneurons and experimental networks of nonlinear electronic circuits evidencing\nsuch a hierarchical behavior. Importantly, our results imply that it is\npossible to infer the degree distribution of a network only from individual\ndynamical measurements.", + "category": "nlin_CD" + }, + { + "text": "Stochastic suspensions of heavy particles: Turbulent suspensions of heavy particles in incompressible flows have gained\nmuch attention in recent years. A large amount of work focused on the impact\nthat the inertia and the dissipative dynamics of the particles have on their\ndynamical and statistical properties. Substantial progress followed from the\nstudy of suspensions in model flows which, although much simpler, reproduce\nmost of the important mechanisms observed in real turbulence. This paper\npresents recent developments made on the relative motion of a pair of particles\nsuspended in time-uncorrelated and spatially self-similar Gaussian flows. This\nreview is complemented by new results. By introducing a time-dependent Stokes\nnumber, it is demonstrated that inertial particle relative dispersion recovers\nasymptotically Richardson's diffusion associated to simple tracers. A\nperturbative (homogeneization) technique is used in the small-Stokes-number\nasymptotics and leads to interpreting first-order corrections to tracer\ndynamics in terms of an effective drift. This expansion implies that the\ncorrelation dimension deficit behaves linearly as a function of the Stokes\nnumber. The validity and the accuracy of this prediction is confirmed by\nnumerical simulations.", + "category": "nlin_CD" + }, + { + "text": "Non-integrability of the dumbbell and point mass problem: This paper discusses a constrained gravitational three-body problem with two\nof the point masses separated by a massless inflexible rod to form a dumbbell.\nThe non-integrability of this system is proven using differential Galois\ntheory.", + "category": "nlin_CD" + }, + { + "text": "Interplay of Delay and multiplexing: Impact on Cluster Synchronization: Communication delays and multiplexing are ubiquitous features of real-world\nnetworked systems. We here introduce a simple model where these two features\nare simultaneously present, and report the rich phe- nomenology which is\nactually due to their interplay on cluster synchronization. A delay in one\nlayer has non trivial impacts on the collective dynamics of the other layers,\nenhancing or suppressing synchronization. At the same time, multiplexing may\nalso enhance cluster synchronization of delayed layers. We elucidate several\nnon trivial (and anti-intuitive) scenarios, which are of interest and potential\napplication in various real-world systems, where introduction of a delay may\nrender synchronization of a layer robust against changes in the properties of\nthe other layers.", + "category": "nlin_CD" + }, + { + "text": "Suppressing noise-induced intensity pulsations in semiconductor lasers\n by means of time-delayed feedback: We investigate the possibility to suppress noise-induced intensity pulsations\n(relaxation oscillations) in semiconductor lasers by means of a time-delayed\nfeedback control scheme. This idea is first studied in a generic normal form\nmodel, where we derive an analytic expression for the mean amplitude of the\noscillations and demonstrate that it can be strongly modulated by varying the\ndelay time. We then investigate the control scheme analytically and numerically\nin a laser model of Lang-Kobayashi type and show that relaxation oscillations\nexcited by noise can be very efficiently suppressed via feedback from a\nFabry-Perot resonator.", + "category": "nlin_CD" + }, + { + "text": "The First Birkhoff Coefficient and the Stability of 2-Periodic Orbits on\n Billiards: In this work we address the question of proving the stability of elliptic\n2-periodic orbits for strictly convex billiards. Eventhough it is part of a\nwidely accepted belief that ellipticity implies stability, classical theorems\nshow that the certainty of stability relies upon more fine conditions. We\npresent a review of the main results and general theorems and describe the\nprocedure to fullfill the supplementary conditions for strictly convex\nbilliards.", + "category": "nlin_CD" + }, + { + "text": "Random matrix description of decaying quantum systems: This contribution describes a statistical model for decaying quantum systems\n(e.g. photo-dissociation or -ionization). It takes the interference between\ndirect and indirect decay processes explicitely into account. The resulting\nexpressions for the partial decay amplitudes and the corresponding cross\nsections may be considered a many-channel many-resonance generalization of\nFano's original work on resonance lineshapes [Phys. Rev 124, 1866 (1961)].\n A statistical (random matrix) model is then introduced. It allows to describe\nchaotic scattering systems with tunable couplings to the decay channels. We\nfocus on the autocorrelation function of the total (photo) cross section, and\nwe find that it depends on the same combination of parameters, as the\nFano-parameter distribution. These combinations are statistical variants of the\none-channel Fano parameter. It is thus possible to study Fano interference\n(i.e. the interference between direct and indirect decay paths) on the basis of\nthe autocorrelation function, and thereby in the regime of overlapping\nresonances. It allows us, to study the Fano interference in the limit of\nstrongly overlapping resonances, where we find a persisting effect on the level\nof the weak localization correction.", + "category": "nlin_CD" + }, + { + "text": "Controlling Chaotic transport on Periodic Surfaces: We uncover and characterize different chaotic transport scenarios on perfect\nperiodic surfaces by controlling the chaotic dynamics of particles subjected to\nperiodic external forces in the absence of a ratchet effect. After identifying\nrelevant {\\it symmetries} of chaotic solutions, analytical estimates in\nparameter space for the occurrence of different transport scenarios are\nprovided and confirmed by numerical simulations. These scenarios are highly\nsensitive to variations of the system's asymmetry parameters, including the\neccentricity of the periodic surface and the direction of dc and ac forces,\nwhich could be useful for particle sorting purposes in those cases where chaos\nis unavoidable.", + "category": "nlin_CD" + }, + { + "text": "Anomalous scaling of passively advected magnetic field in the presence\n of strong anisotropy: Inertial-range scaling behavior of high-order (up to order N=51) structure\nfunctions of a passively advected vector field has been analyzed in the\nframework of the rapid-change model with strong small-scale anisotropy with the\naid of the renormalization group and the operator-product expansion. It has\nbeen shown that in inertial range the leading terms of the structure functions\nare coordinate independent, but powerlike corrections appear with the same\nanomalous scaling exponents as for the passively advected scalar field. These\nexponents depend on anisotropy parameters in such a way that a specific\nhierarchy related to the degree of anisotropy is observed. Deviations from\npower-law behavior like oscillations or logarithmic behavior in the corrections\nto structure functions have not been found.", + "category": "nlin_CD" + }, + { + "text": "Knowledge-Based Learning of Nonlinear Dynamics and Chaos: Extracting predictive models from nonlinear systems is a central task in\nscientific machine learning. One key problem is the reconciliation between\nmodern data-driven approaches and first principles. Despite rapid advances in\nmachine learning techniques, embedding domain knowledge into data-driven models\nremains a challenge. In this work, we present a universal learning framework\nfor extracting predictive models from nonlinear systems based on observations.\nOur framework can readily incorporate first principle knowledge because it\nnaturally models nonlinear systems as continuous-time systems. This both\nimproves the extracted models' extrapolation power and reduces the amount of\ndata needed for training. In addition, our framework has the advantages of\nrobustness to observational noise and applicability to irregularly sampled\ndata. We demonstrate the effectiveness of our scheme by learning predictive\nmodels for a wide variety of systems including a stiff Van der Pol oscillator,\nthe Lorenz system, and the Kuramoto-Sivashinsky equation. For the Lorenz\nsystem, different types of domain knowledge are incorporated to demonstrate the\nstrength of knowledge embedding in data-driven system identification.", + "category": "nlin_CD" + }, + { + "text": "Periodic orbits, basins of attraction and chaotic beats in two coupled\n Kerr oscillators: Kerr oscillators are model systems which have practical applications in\nnonlinear optics. Optical Kerr effect i.e. interaction of optical waves with\nnonlinear medium with polarizability $\\chi^{(3)}$ is the basic phenomenon\nneeded to explain for example the process of light transmission in fibers and\noptical couplers. In this paper we analyze the two Kerr oscillators coupler and\nwe show that there is a possibility to control the dynamics of this system,\nespecially by switching its dynamics from periodic to chaotic motion and vice\nversa. Moreover the switching between two different stable periodic states is\ninvestigated. The stability of the system is described by the so-called maps of\nLyapunov exponents in parametric spaces. Comparison of basins of attractions\nbetween two Kerr couplers and a single Kerr system is also presented.", + "category": "nlin_CD" + }, + { + "text": "A route to chaos in the Boros-Moll map: The Boros-Moll map appears as a subsystem of a Landen transformation\nassociated to certain rational integrals and its dynamics is related to the\nconvergence of them. In the paper, we study the dynamics of a one-parameter\nfamily of maps which unfolds the Boros-Moll one, showing that the existence of\nan unbounded invariant chaotic region in the Boros-Moll map is a peculiar\nfeature within the family. We relate this singularity with a specific property\nof the critical lines that occurs only for this special case. In particular, we\nexplain how the unbounded chaotic region in the Boros-Moll map appears. Special\nattention is devoted to explain the main contact/homoclinic bifurcations that\noccur in the family. We also report some other bifurcation phenomena that\nappear in the considered unfolding.", + "category": "nlin_CD" + }, + { + "text": "A Unified Approach to Attractor Reconstruction: In the analysis of complex, nonlinear time series, scientists in a variety of\ndisciplines have relied on a time delayed embedding of their data, i.e.\nattractor reconstruction. The process has focused primarily on heuristic and\nempirical arguments for selection of the key embedding parameters, delay and\nembedding dimension. This approach has left several long-standing, but common\nproblems unresolved in which the standard approaches produce inferior results\nor give no guidance at all. We view the current reconstruction process as\nunnecessarily broken into separate problems. We propose an alternative approach\nthat views the problem of choosing all embedding parameters as being one and\nthe same problem addressable using a single statistical test formulated\ndirectly from the reconstruction theorems. This allows for varying time delays\nappropriate to the data and simultaneously helps decide on embedding dimension.\nA second new statistic, undersampling, acts as a check against overly long time\ndelays and overly large embedding dimension. Our approach is more flexible than\nthose currently used, but is more directly connected with the mathematical\nrequirements of embedding. In addition, the statistics developed guide the user\nby allowing optimization and warning when embedding parameters are chosen\nbeyond what the data can support. We demonstrate our approach on uni- and\nmultivariate data, data possessing multiple time scales, and chaotic data. This\nunified approach resolves all the main issues in attractor reconstruction.", + "category": "nlin_CD" + }, + { + "text": "Synchronous Behavior of Coupled Systems with Discrete Time: The dynamics of one-way coupled systems with discrete time is considered. The\nbehavior of the coupled logistic maps is compared to the dynamics of maps\nobtained using the Poincare sectioning procedure applied to the coupled\ncontinuous-time systems in the phase synchronization regime. The behavior\n(previously considered as asynchronous) of the coupled maps that appears when\nthe complete synchronization regime is broken as the coupling parameter\ndecreases, corresponds to the phase synchronization of flow systems, and should\nbe considered as a synchronous regime. A quantitative measure of the degree of\nsynchronism for the interacting systems with discrete time is proposed.", + "category": "nlin_CD" + }, + { + "text": "A review of linear response theory for general differentiable dynamical\n systems: The classical theory of linear response applies to statistical mechanics\nclose to equilibrium. Away from equilibrium, one may describe the microscopic\ntime evolution by a general differentiable dynamical system, identify\nnonequilibrium steady states (NESS), and study how these vary under\nperturbations of the dynamics. Remarkably, it turns out that for uniformly\nhyperbolic dynamical systems (those satisfying the \"chaotic hypothesis\"), the\nlinear response away from equilibrium is very similar to the linear response\nclose to equilibrium: the Kramers-Kronig dispersion relations hold, and the\nfluctuation-dispersion theorem survives in a modified form (which takes into\naccount the oscillations around the \"attractor\" corresponding to the NESS). If\nthe chaotic hypothesis does not hold, two new phenomena may arise. The first is\na violation of linear response in the sense that the NESS does not depend\ndifferentiably on parameters (but this nondifferentiability may be hard to see\nexperimentally). The second phenomenon is a violation of the dispersion\nrelations: the susceptibility has singularities in the upper half complex\nplane. These \"acausal\" singularities are actually due to \"energy\nnonconservation\": for a small periodic perturbation of the system, the\namplitude of the linear response is arbitrarily large. This means that the NESS\nof the dynamical system under study is not \"inert\" but can give energy to the\noutside world. An \"active\" NESS of this sort is very different from an\nequilibrium state, and it would be interesting to see what happens for active\nstates to the Gallavotti-Cohen fluctuation theorem.", + "category": "nlin_CD" + }, + { + "text": "Chaos and Stochastic Models in Physics: Ontic and Epistemic Aspects: There is a persistent confusion about determinism and predictability. In\nspite of the opinions of some eminent philosophers (e.g., Popper), it is\npossible to understand that the two concepts are completely unrelated. In few\nwords we can say that determinism is ontic and has to do with how Nature\nbehaves, while predictability is epistemic and is related to what the human\nbeings are able to compute. An analysis of the Lyapunov exponents and the\nKolmogorov-Sinai entropy shows how deterministic chaos, although with an\nepistemic character, is non subjective at all. This should clarify the role and\ncontent of stochastic models in the description of the physical world.", + "category": "nlin_CD" + }, + { + "text": "Model-free measure of coupling from embedding principle: A model-free measure of coupling between dynamical variables is built from\ntime series embedding principle. The approach described does not require a\nmathematical form for the dynamics to be assumed. The approach also does not\nrequire density estimation which is an intractable problem in high dimensions.\nThe measure has strict asymptotic bounds and is robust to noise. The proposed\napproach is used to demonstrate coupling between complex time series from the\nfinance world.", + "category": "nlin_CD" + }, + { + "text": "A quasi-periodic route to chaos in a parametrically driven nonlinear\n medium: Small-sized systems exhibit a finite number of routes to chaos. However, in\nextended systems, not all routes to complex spatiotemporal behavior have been\nfully explored. Starting from the sine-Gordon model of parametrically driven\nchain of damped nonlinear oscillators, we investigate a route to spatiotemporal\nchaos emerging from standing waves. The route from the stationary to the\nchaotic state proceeds through quasiperiodic dynamics. The standing wave\nundergoes the onset of oscillatory instability, which subsequently exhibits a\ndifferent critical frequency, from which the complexity originates. A suitable\namplitude equation, valid close to the parametric resonance, makes it possible\nto produce universe results. The respective phase-space structure and\nbifurcation diagrams are produced in a numerical form. We characterize the\nrelevant dynamical regimes by means of the largest Lyapunov exponent, the power\nspectrum, and the evolution of the total intensity of the wave field.", + "category": "nlin_CD" + }, + { + "text": "Jacobian deformation ellipsoid and Lyapunov stability analysis revisited: The stability analysis introduced by Lyapunov and extended by Oseledec is an\nexcellent tool to describe the character of nonlinear n-dimensional flows by n\nglobal exponents if these flows are stable in time. However, there are two main\nshortcomings: (a) The local exponents fail to indicate the origin of\ninstability where trajectories start to diverge. Instead, their time evolution\ncontains a much stronger chaos than the trajectories, which is only eliminated\nby integrating over a long time. Therefore, shorter time intervals cannot be\ncharacterized correctly, which would be essential to analyse changes of chaotic\ncharacter as in transients. (b) Moreover, although Oseledec uses an n\ndimensional sphere around a point x to be transformed into an n dimensional\nellipse in first order, this local ellipse has yet not been evaluated. The aim\nof this contribution is to eliminate these two shortcomings. Problem (a)\ndisappears if the Oseledec method is replaced by a frame with a 'constraint' as\nperformed by Rateitschak and Klages (RK) [Phys. Rev. E 65 036209 (2002)]. The\nreasons why this method is better will be illustrated by comparing different\nsystems. In order to analyze shorter time intervals, integrals between\nconsecutive Poincare points will be evaluated. The local problems (b) will be\nsolved analytically by introducing the symmetric 'Jacobian deformation\nellipsoid' and its orthogonal submatrix, which enable to search in the full\nphase space for extreme local separation exponents. These are close to the RK\nexponents but need no time integration of the RK frame. Finally, four sets of\nlocal exponents are compared: Oseledec frame, RK frame, Jacobian deformation\nellipsoid and its orthogonal submatrix.", + "category": "nlin_CD" + }, + { + "text": "Many Roads to Synchrony: Natural Time Scales and Their Algorithms: We consider two important time scales---the Markov and cryptic orders---that\nmonitor how an observer synchronizes to a finitary stochastic process. We show\nhow to compute these orders exactly and that they are most efficiently\ncalculated from the epsilon-machine, a process's minimal unifilar model.\nSurprisingly, though the Markov order is a basic concept from stochastic\nprocess theory, it is not a probabilistic property of a process. Rather, it is\na topological property and, moreover, it is not computable from any\nfinite-state model other than the epsilon-machine. Via an exhaustive survey, we\nclose by demonstrating that infinite Markov and infinite cryptic orders are a\ndominant feature in the space of finite-memory processes. We draw out the roles\nplayed in statistical mechanical spin systems by these two complementary length\nscales.", + "category": "nlin_CD" + }, + { + "text": "Transition from anticipatory to lag synchronization via complete\n synchronization in time-delay systems: The existence of anticipatory, complete and lag synchronization in a single\nsystem having two different time-delays, that is feedback delay $\\tau_1$ and\ncoupling delay $\\tau_2$, is identified. The transition from anticipatory to\ncomplete synchronization and from complete to lag synchronization as a function\nof coupling delay $\\tau_2$ with suitable stability condition is discussed. The\nexistence of anticipatory and lag synchronization is characterized both by the\nminimum of similarity function and the transition from on-off intermittency to\nperiodic structure in laminar phase distribution.", + "category": "nlin_CD" + }, + { + "text": "Synchronization of hypernetworks of coupled dynamical systems: We consider synchronization of coupled dynamical systems when different types\nof interactions are simultaneously present. We assume that a set of dynamical\nsystems are coupled through the connections of two or more distinct networks\n(each of which corresponds to a distinct type of interaction), and we refer to\nsuch a system as a hypernetwork. Applications include neural networks formed of\nboth electrical gap junctions and chemical synapses, the coordinated motion of\nshoals of fishes communicating through both vision and flow sensing, and\nhypernetworks of coupled chaotic oscillators. We first analyze the case of a\nhypernetwork formed of $m=2$ networks. We look for necessary and sufficient\nconditions for synchronization. We attempt at reducing the linear stability\nproblem in a master stability function form, i.e., at decoupling the effects of\nthe coupling functions from the structure of the networks. Unfortunately, we\nare unable to obtain a reduction in a master stability function form for the\ngeneral case. However, we show that such a reduction is possible in three cases\nof interest: (i) the Laplacian matrices associated with the two networks\ncommute; (ii) one of the two networks is unweighted and fully connected; (iii)\none of the two networks is such that the coupling strength from node $i$ to\nnode $j$ is a function of $j$ but not of $i$. Furthermore, we define a class of\nnetworks such that if either one of the two coupling networks belongs to this\nclass, the reduction can be obtained independently of the other network. As an\nexample of interest, we study synchronization of a neural hypernetwork for\nwhich the connections can be either chemical synapses or electrical gap\njunctions. We propose a generalization of our stability results to the case of\nhypernetworks formed of $m\\geq 2$ networks.", + "category": "nlin_CD" + }, + { + "text": "Parameter Mismatches, Chaos Synchronization and Fast Dynamic Logic Gates: By using chaos synchronization between non-identical multiple time delay\nsemiconductor lasers with optoelectronic feedbacks, we demonstrate numerically\nhow fast dynamic logic gates can be constructed. The results may be helpful to\nobtain a computational hardware with reconfigurable properties.", + "category": "nlin_CD" + }, + { + "text": "Application of largest Lyapunov exponent analysis on the studies of\n dynamics under external forces: Dynamics of driven dissipative Frenkel-Kontorova model is examined by using\nlargest Lyapunov exponent computational technique. Obtained results show that\nbesides the usual way where behavior of the system in the presence of external\nforces is studied by analyzing its dynamical response function, the largest\nLyapunov exponent analysis can represent a very convenient tool to examine\nsystem dynamics. In the dc driven systems, the critical depinning force for\nparticular structure could be estimated by computing the largest Lyapunov\nexponent. In the dc+ac driven systems, if the substrate potential is the\nstandard sinusoidal one, calculation of the largest Lyapunov exponent offers a\nmore sensitive way to detect the presence of Shapiro steps. When the amplitude\nof the ac force is varied the behavior of the largest Lyapunov exponent in the\npinned regime completely reflects the behavior of Shapiro steps and the\ncritical depinning force, in particular, it represents the mirror image of the\namplitude dependence of critical depinning force. This points out an advantage\nof this technique since by calculating the largest Lyapunov exponent in the\npinned regime we can get an insight into the dynamics of the system when\ndriving forces are applied.", + "category": "nlin_CD" + }, + { + "text": "Vibrational resonance in groundwater-dependent plant ecosystems: We report the phenomenon of vibrational resonance in a single species and a\ntwo species models of groundwater-dependent plant ecosystems with a biharmonic\noscillation (with two widely different frequencies \\omega and \\Omega, \\Omega >>\n\\omega) of the water table depth. In these two systems, the response amplitude\nof the species biomass shows multiple resonances with different mechanisms. The\nresonance occurs at both low- and high-frequencies of the biharmonic force. In\nthe single species bistable system, the resonance occurs at discrete values of\nthe amplitude g of the high-frequency component of the water table.\nFurthermore, the best synchronization of biomass and its carrying capacity with\nthe biharmonic force occurs at the resonance. In the two species excitable and\ntime-delay model, the response amplitude (Q) profile shows several plateau\nregions of resonance, where the period of evolution of the species biomass\nremains the same and the value of Q is inversely proportional to it. The\nresponse amplitude is highly sensitive to the time-delay parameter \\tau and\nshows two distinct sequences of resonance intervals with a decreasing amplitude\nwith \\tau.", + "category": "nlin_CD" + }, + { + "text": "Anomalous transport and observable average in the standard map: The distribution of finite time observable averages and transport in low\ndimensional Hamiltonian systems is studied. Finite time observable average\ndistributions are computed, from which an exponent $\\alpha$ characteristic of\nhow the maximum of the distributions scales with time is extracted. To link\nthis exponent to transport properties, the characteristic exponent $\\mu(q)$ of\nthe time evolution of the different moments of order $q$ related to transport\nare computed. As a testbed for our study the standard map is used. The\nstochasticity parameter $K$ is chosen so that either phase space is mixed with\na chaotic sea and islands of stability or with only a chaotic sea. Our\nobservations lead to a proposition of a law relating the slope in $q=0$ of the\nfunction $\\mu(q)$ with the exponent $\\alpha$.", + "category": "nlin_CD" + }, + { + "text": "Kneadings, Symbolic Dynamics and Painting Lorenz Chaos. A Tutorial: A new computational technique based on the symbolic description utilizing\nkneading invariants is proposed and verified for explorations of dynamical and\nparametric chaos in a few exemplary systems with the Lorenz attractor. The\ntechnique allows for uncovering the stunning complexity and universality of\nbi-parametric structures and detect their organizing centers - codimension-two\nT-points and separating saddles in the kneading-based scans of the iconic\nLorenz equation from hydrodynamics, a normal model from mathematics, and a\nlaser model from nonlinear optics.", + "category": "nlin_CD" + }, + { + "text": "Hysteresis Models of Dynamic Mode Atomic Force Microscopes: Analysis and\n Identification: A new class of models based on hysteresis functions is developed to describe\natomic force microscopes operating in dynamic mode. Such models are able to\naccount for dissipative phenomena in the tip-sample interaction which are\npeculiar of this operation mode. The model analysis, which can be pursued using\nfrequency domain techniques, provides a clear insight of specific nonlinear\nbehaviours. Experiments show good agreement with the identified models.", + "category": "nlin_CD" + }, + { + "text": "Decoding Information by Following Parameter Modulation With Parameter\n Adaptive Control: It has been proposed to realize secure communication using chaotic\nsynchronization via transmission of binary message encoded by parameter\nmodulation in the chaotic system. This paper considers the use of parameter\nadaptive control techniques to extract the message, based on the assumptions\nthat we know the equation form of the chaotic system in the transmitter but do\nnot have access to the precise values of the parameters which are kept secret\nas a secure set. In the case that a synchronizing system can be constructed\nusing parameter adaptive control by the transmitted signal and the\nsynchronization is robust to parameter mismatches, the parameter modulation can\nbe revealed and the message decoded without resorting to exact parameter values\nin the secure set. A practical local Lyapunov function method for designing\nparameter adaptive control rules based on originally synchronized systems is\npresented.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of Rotator Chain with Dissipative Boundary: We study the deterministic dynamics of rotator chain that subjected to purely\nmechanical driving on the boundary by stability analysis and numerical\nsimulation. Globally synchronous rotation, clustered synchronous rotation, and\nsplit synchronous rotation states are identified. In particular, we find that\nthe single-peaked variance distribution of angular momenta is the consequence\nof the deterministic dynamics. As a result, the operational definition of\ntemperature used in the previous studies on rotator chain should be revisited.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling: Small lattices of $N$ nearest neighbor coupled excitable FitzHugh-Nagumo\nsystems, with time-delayed coupling are studied, and compared with systems of\nFitzHugh-Nagumo oscillators with the same delayed coupling. Bifurcations of\nequilibria in N=2 case are studied analytically, and it is then numerically\nconfirmed that the same bifurcations are relevant for the dynamics in the case\n$N>2$. Bifurcations found include inverse and direct Hopf and fold limit cycle\nbifurcations. Typical dynamics for different small time-lags and coupling\nintensities could be excitable with a single globally stable equilibrium,\nasymptotic oscillatory with symmetric limit cycle, bi-stable with stable\nequilibrium and a symmetric limit cycle, and again coherent oscillatory but\nnon-symmetric and phase-shifted. For an intermediate range of time-lags inverse\nsub-critical Hopf and fold limit cycle bifurcations lead to the phenomenon of\noscillator death. The phenomenon does not occur in the case of FitzHugh-Nagumo\noscillators with the same type of coupling.", + "category": "nlin_CD" + }, + { + "text": "Non-identical multiplexing promotes chimera states: We present the emergence of chimeras, a state referring to coexistence of\npartly coherent, partly incoherent dynamics in networks of identical\noscillators, in a multiplex network consisting of two non-identical layers\nwhich are interconnected. We demonstrate that the parameter range displaying\nthe chimera state in the homogeneous first layer of the multiplex networks can\nbe tuned by changing the link density or connection architecture of the same\nnodes in the second layer. We focus on the impact of the interconnected second\nlayer on the enlargement or shrinking of the coupling regime for which chimeras\nare displayed in the homogeneous first layer. We find that a denser homogeneous\nsecond layer promotes chimera in a sparse first layer, where chimeras do not\noccur in isolation. Furthermore, while a dense connection density is required\nfor the second layer if it is homogeneous, this is not true if the second layer\nis inhomogeneous. We demonstrate that a sparse inhomogeneous second layer which\nis common in real-world complex systems can promote chimera states in a sparse\nhomogeneous first layer.", + "category": "nlin_CD" + }, + { + "text": "Hydrodynamic superradiance in wave-mediated cooperative tunneling: Superradiance and subradiance occur in quantum optics when the emission rate\nof photons from multiple atoms is enhanced and diminished, respectively, owing\nto interaction between neighboring atoms. We here demonstrate a classical\nanalog thereof in a theoretical model of droplets walking on a vibrating bath.\nTwo droplets are confined to identical two-level systems, a pair of wells\nbetween which the drops may tunnel, joined by an intervening coupling cavity.\nThe resulting classical superradiance is rationalized in terms of the system's\nnon-Markovian, pilot-wave dynamics.", + "category": "nlin_CD" + }, + { + "text": "Random Wandering Around Homoclinic-like Manifolds in Symplectic Map\n Chain: We present a method to construct a symplecticity preserving renormalization\ngroup map of a chain of weakly nonlinear symplectic maps and obtain a general\nreduced symplectic map describing its long-time behaviour. It is found that the\nmodulational instability in the reduced map triggers random wandering of orbits\naround some homoclinic-like manifolds, which is understood as the Bernoulli\nshifts.", + "category": "nlin_CD" + }, + { + "text": "Almost Periodicity in Chaos: Periodicity plays a significant role in the chaos theory from the beginning\nsince the skeleton of chaos can consist of infinitely many unstable periodic\nmotions. This is true for chaos in the sense of Devaney [1], Li-Yorke [2] and\nthe one obtained through period-doubling cascade [3]. Countable number of\nperiodic orbits exist in any neighborhood of a structurally stable Poincar\\'{e}\nhomoclinic orbit, which can be considered as a criterion for the presence of\ncomplex dynamics [4]-[6]. It was certified by Shilnikov [7] and Seifert [8]\nthat it is possible to replace periodic solutions by Poisson stable or almost\nperiodic motions in a chaotic attractor. Despite the fact that the idea of\nreplacing periodic solutions by other types of regular motions is attractive,\nvery few results have been obtained on the subject. The present study\ncontributes to the chaos theory in that direction.\n In this paper, we take into account chaos both through a cascade of almost\nperiodic solutions and in the sense of Li-Yorke such that the original Li-Yorke\ndefinition is modified by replacing infinitely many periodic motions with\nalmost periodic ones, which are separated from the motions of the scrambled\nset. The theoretical results are valid for systems with arbitrary high\ndimensions. Formation of the chaos is exemplified by means of unidirectionally\ncoupled Duffing oscillators. The controllability of the extended chaos is\ndemonstrated numerically by means of the Ott-Grebogi-Yorke [9] control\ntechnique. In particular, the stabilization of tori is illustrated.", + "category": "nlin_CD" + }, + { + "text": "Linear and fractal diffusion coefficients in a family of one dimensional\n chaotic maps: We analyse deterministic diffusion in a simple, one-dimensional setting\nconsisting of a family of four parameter dependent, chaotic maps defined over\nthe real line. When iterated under these maps, a probability density function\nspreads out and one can define a diffusion coefficient. We look at how the\ndiffusion coefficient varies across the family of maps and under parameter\nvariation. Using a technique by which Taylor-Green-Kubo formulae are evaluated\nin terms of generalised Takagi functions, we derive exact, fully analytical\nexpressions for the diffusion coefficients. Typically, for simple maps these\nquantities are fractal functions of control parameters. However, our family of\nfour maps exhibits both fractal and linear behavior. We explain these different\nstructures by looking at the topology of the Markov partitions and the ergodic\nproperties of the maps.", + "category": "nlin_CD" + }, + { + "text": "Power spectrum analysis and missing level statistics of microwave graphs\n with violated time reversal invariance: We present experimental studies of the power spectrum and other fluctuation\nproperties in the spectra of microwave networks simulating chaotic quantum\ngraphs with violated time reversal in- variance. On the basis of our data sets\nwe demonstrate that the power spectrum in combination with other long-range and\nalso short-range spectral fluctuations provides a powerful tool for the\nidentification of the symmetries and the determination of the fraction of\nmissing levels. Such a pro- cedure is indispensable for the evaluation of the\nfluctuation properties in the spectra of real physical systems like, e.g.,\nnuclei or molecules, where one has to deal with the problem of missing levels.", + "category": "nlin_CD" + }, + { + "text": "Boundary crisis and suppression of Fermi acceleration in a dissipative\n two dimensional non-integrable time-dependent billiard: Some dynamical properties for a dissipative time-dependent oval-shaped\nbilliard are studied. The system is described in terms of a four-dimensional\nnonlinear mapping. Dissipation is introduced via inelastic collisions of the\nparticle with the boundary, thus implying that the particle has a fractional\nloss of energy upon collision. The dissipation causes profound modifications in\nthe dynamics of the particle as well as in the phase space of the non\ndissipative system. In particular, inelastic collisions can be assumed as an\nefficient mechanism to suppress Fermi acceleration of the particle. The\ndissipation also creates attractors in the system, including chaotic. We show\nthat a slightly modification of the intensity of the damping coefficient yields\na drastic and sudden destruction of the chaotic attractor, thus leading the\nsystem to experience a boundary crisis. We have characterized such a boundary\ncrisis via a collision of the chaotic attractor with its own basin of\nattraction and confirmed that inelastic collisions do indeed suppress Fermi\nacceleration in two-dimensional time dependent billiards.", + "category": "nlin_CD" + }, + { + "text": "An Analytical Study on the Synchronization of Strange Non-Chaotic\n Attractors: In this paper we present an analytical study on the synchronization dynamics\nobserved in unidirectionally-coupled quasiperiodically-forced systems that\nexhibit Strange Non-chaotic Attractors (SNA) in their dynamics. The SNA\ndynamics observed in the uncoupled system is studied analytically through phase\nportraits and poincare maps. A difference system is obtained by coupling the\nstate equations of similar piecewise linear regions of the drive and response\nsystems. The mechanism of synchronization of the coupled system is realized\nthrough the bifurcation of the eigenvalues in one of the piecewise linear\nregions of the difference system. The analytical solutions obtained for the\nnormalized state equations in each piecewise linear region of the difference\nsystem has been used to explain the synchronization dynamics though phase\nportraits and timeseries analysis. The stability of the synchronized state is\nconfirmed through the Master Stability Function. An explicit analytical\nsolution explaining the synchronization of SNAs is reported in the literature\nfor the first time.", + "category": "nlin_CD" + }, + { + "text": "Impulse-induced optimum control of chaos in dissipative driven systems: Taming chaos arising from dissipative non-autonomous nonlinear systems by\napplying additional harmonic excitations is a reliable and widely used\nprocedure nowadays. But the suppressory effectiveness of generic non-harmonic\nperiodic excitations continues to be a significant challenge both to our\ntheoretical understanding and in practical applications. Here we show how the\neffectiveness of generic suppressory excitations is optimally enhanced when the\nimpulse transmitted by them (time integral over two consecutive zeros) is\njudiciously controlled in a not obvious way. This is demonstrated\nexperimentally by means of an analog version of a universal model, and\nconfirmed numerically by simulations of such a damped driven system including\nthe presence of noise. Our theoretical analysis shows that the controlling\neffect of varying the impulse is due to a correlative variation of the energy\ntransmitted by the suppressory excitation.", + "category": "nlin_CD" + }, + { + "text": "Few-Freedom Turbulence: The results of numerical experiments on the structure of chaotic attractors\nin the Khalatnikov - Kroyter model of two freedoms are presented. This model\nwas developed for a qualitative description of the wave turbulence of the\nsecond sound in helium. The attractor dimension, size, and the maximal Lyapunov\nexponent in dependence on the single dimensionless parameter $F$ of the model\nare found and discussed. The principal parameter $F$ is similar to the Reynolds\nnumber in hydrodynamic turbulence. We were able to discern four different\nattractors characterized by a specific critical value of the parameter\n($F=F_{cr}$), such that the attractor exists for $F>F_{cr}$ only. A simple\nempirical relation for this dependence on the argument ($F-F_{cr}$) is\npresented which turns out to be universal for different attractors with respect\nto the dimension and dimensionless Lyapunov exponents. Yet, it differs as to\nthe size of attractor. In the main region of our studies the dependence of all\ndimensionless characteristics of the chaotic attractor on parameter $F$ is very\nslow (logarithmic) which is qualitatively different as compared to that of a\nmulti-freedom attractor, e.g., in hydrodynamic turbulence (a power law).\nHowever, at very large $F\\sim 10^7$ the transition to a power-law dependence\nhas been finally found, similar to the multi-freedom attractor. Some unsolved\nproblems and open questions are also discussed.", + "category": "nlin_CD" + }, + { + "text": "Bailout Embeddings, Targeting of KAM Orbits, and the Control of\n Hamiltonian Chaos: We present a novel technique, which we term bailout embedding, that can be\nused to target orbits having particular properties out of all orbits in a flow\nor map. We explicitly construct a bailout embedding for Hamiltonian systems so\nas to target KAM orbits. We show how the bailout dynamics is able to lock onto\nextremely small KAM islands in an ergodic sea.", + "category": "nlin_CD" + }, + { + "text": "Lorenz cycle for the Lorenz attractor: In this note we study energetics of Lorenz-63 system through its Lie-Poisson\nstructure.", + "category": "nlin_CD" + }, + { + "text": "Designer dynamics through chaotic traps: Controlling complex behavior in\n driven nonlinear systems: Control schemes for dynamical systems typically involve stabilizing unstable\nperiodic orbits. In this paper we introduce a new paradigm of control that\ninvolves `trapping' the dynamics arbitrarily close to any desired trajectory.\nThis is achieved by a state-dependent dynamical selection of the input signal\napplied to the driven nonlinear system. An emergent property of the trapping\nprocess is that the signal changes in a chaotic sequence: a manifestation of\nchaos-induced order. The simplicity of the control scheme makes it easily\nimplementable in experimental systems.", + "category": "nlin_CD" + }, + { + "text": "Synchronization in discrete-time networks with general pairwise coupling: We consider complete synchronization of identical maps coupled through a\ngeneral interaction function and in a general network topology where the edges\nmay be directed and may carry both positive and negative weights. We define\nmixed transverse exponents and derive sufficient conditions for local complete\nsynchronization. The general non-diffusive coupling scheme can lead to new\nsynchronous behavior, in networks of identical units, that cannot be produced\nby single units in isolation. In particular, we show that synchronous chaos can\nemerge in networks of simple units. Conversely, in networks of chaotic units\nsimple synchronous dynamics can emerge; that is, chaos can be suppressed\nthrough synchrony.", + "category": "nlin_CD" + }, + { + "text": "Non-permanent form solutions in the Hamiltonian formulation of surface\n water waves: Using the KAM method, we exhibit some solutions of a finite-dimensional\napproximation of the Zakharov Hamiltonian formulation of gravity water waves,\nwhich are spatially periodic, quasi-periodic in time, and not permanent form\ntravelling waves. For this Hamiltonian, which is the total energy of the waves,\nthe canonical variables are some complex quantities an and a*n, which are\nlinear combinations of the Fourier components of the free surface elevation and\nthe velocity potential evaluated at the surface. We expose the method for the\ncase of a system with a finite number of degrees of freedom, the Zufiria model,\nwith only 3 modes interacting.", + "category": "nlin_CD" + }, + { + "text": "About universality of lifetime statistics in quantum chaotic scattering: The statistics of the resonance widths and the behavior of the survival\nprobability is studied in a particular model of quantum chaotic scattering (a\nparticle in a periodic potential subject to static and time-periodic forces)\nintroduced earlier in Ref.[5,6]. The coarse-grained distribution of the\nresonance widths is shown to be in good agreement with the prediction of Random\nMatrix Theory (RMT). The behavior of the survival probability shows, however,\nsome deviation from RMT.", + "category": "nlin_CD" + }, + { + "text": "Spatial Patterns in Chemically and Biologically Reacting Flows: We present here a number of processes, inspired by concepts in Nonlinear\nDynamics such as chaotic advection and excitability, that can be useful to\nunderstand generic behaviors in chemical or biological systems in fluid flows.\nEmphasis is put on the description of observed plankton patchiness in the sea.\nThe linearly decaying tracer, and excitable kinetics in a chaotic flow are\nmainly the models described. Finally, some warnings are given about the\ndifficulties in modeling discrete individuals (such as planktonic organisms) in\nterms of continuous concentration fields.", + "category": "nlin_CD" + }, + { + "text": "A Diagrammatic Representation of Phase Portraits and Bifurcation\n Diagrams of Two-Dimensional Dynamical Systems: We treat the problem of characterizing in a systematic way the qualitative\nfeatures of two-dimensional dynamical systems. To that end, we construct a\nrepresentation of the topological features of phase portraits by means of\ndiagrams that discard their quantitative information. All codimension 1\nbifurcations are naturally embodied in the possible ways of transitioning\nsmoothly between diagrams. We introduce a representation of bifurcation curves\nin parameter space that guides the proposition of bifurcation diagrams\ncompatible with partial information about the system.", + "category": "nlin_CD" + }, + { + "text": "Lyapunov exponent and natural invariant density determination of chaotic\n maps: An iterative maximum entropy ansatz: We apply the maximum entropy principle to construct the natural invariant\ndensity and Lyapunov exponent of one-dimensional chaotic maps. Using a novel\nfunction reconstruction technique that is based on the solution of Hausdorff\nmoment problem via maximizing Shannon entropy, we estimate the invariant\ndensity and the Lyapunov exponent of nonlinear maps in one-dimension from a\nknowledge of finite number of moments. The accuracy and the stability of the\nalgorithm are illustrated by comparing our results to a number of nonlinear\nmaps for which the exact analytical results are available. Furthermore, we also\nconsider a very complex example for which no exact analytical result for\ninvariant density is available. A comparison of our results to those available\nin the literature is also discussed.", + "category": "nlin_CD" + }, + { + "text": "Stability of helical tubes conveying fluid: We study the linear stability of elastic collapsible tubes conveying fluid,\nwhen the equilibrium configuration of the tube is helical. A particular case of\nsuch tubes, commonly encountered in applications, is represented by quarter- or\nsemi-circular tubular joints used at pipe's turning points. The stability\ntheory for pipes with non-straight equilibrium configurations, especially for\ncollapsible tubes, allowing dynamical change of the cross-section, has been\nelusive as it is difficult to accurately develop the dynamic description via\ntraditional methods. We develop a methodology for studying the\nthree-dimensional dynamics of collapsible tubes based on the geometric\nvariational approach. We show that the linear stability theory based on this\napproach allows for a complete treatment for arbitrary three-dimensional\nhelical configurations of collapsible tubes by reduction to an equation with\nconstant coefficients. We discuss new results on stability loss of straight\ntubes caused by the cross-sectional area change. Finally, we develop a\nnumerical algorithm for computation of the linear stability using our theory\nand present the results of numerical studies for both straight and helical\ntubes.", + "category": "nlin_CD" + }, + { + "text": "First experimental observation of generalized synchronization phenomena\n in microwave oscillators: In this Letter we report for the first time on the experimental observation\nof the generalized synchronization regime in the microwave electronic systems,\nnamely, in the multicavity klystron generators. A new approach devoted to the\ngeneralized synchronization detection has been developed. The experimental\nobservations are in the excellent agreement with the results of numerical\nsimulation. The observed phenomena gives a strong potential for new\napplications requiring microwave chaotic signals.", + "category": "nlin_CD" + }, + { + "text": "Dimensional collapse and fractal attractors of a system with fluctuating\n delay times: A frequently encountered situation in the study of delay systems is that the\nlength of the delay time changes with time, which is of relevance in many\nfields such as optics, mechanical machining, biology or physiology. A\ncharacteristic feature of such systems is that the dimension of the system\ndynamics collapses due to the fluctuations of delay times. In consequence, the\nsupport of the long-trajectory attractors of this kind of systems is found\nbeing fractal in contrast to the fuzzy attractors in most random systems.", + "category": "nlin_CD" + }, + { + "text": "Strong effect of weak diffusion on scalar turbulence at large scales: Passive scalar turbulence forced steadily is characterized by the velocity\ncorrelation scale, $L$, injection scale, $l$, and diffusive scale, $r_d$. The\nscales are well separated if the diffusivity is small, $r_d\\ll l,L$, and one\nnormally says that effects of diffusion are confined to smaller scales, $r\\ll\nr_d$. However, if the velocity is single scale one finds that a weak dependence\nof the scalar correlations on the molecular diffusivity persists to even larger\nscales, e.g. $l\\gg r\\gg r_d$ \\cite{95BCKL}. We consider the case of $L\\gg l$\nand report a counter-intuitive result -- the emergence of a new range of large\nscales, $L\\gg r\\gg l^2/r_d$, where the diffusivity shows a strong effect on\nscalar correlations.", + "category": "nlin_CD" + }, + { + "text": "Poincar\u00e9 chaos and unpredictable functions: The results of this study are continuation of the research of Poincar\\'e\nchaos initiated in papers (Akhmet M, Fen MO. Commun Nonlinear Sci Numer Simulat\n2016;40:1-5; Akhmet M, Fen MO. Turk J Math, doi:10.3906/mat-1603-51, accepted).\nWe focus on the construction of an unpredictable function, continuous on the\nreal axis. As auxiliary results, unpredictable orbits for the symbolic dynamics\nand the logistic map are obtained. By shaping the unpredictable function as\nwell as Poisson function we have performed the first step in the development of\nthe theory of unpredictable solutions for differential and discrete equations.\nThe results are preliminary ones for deep analysis of chaos existence in\ndifferential and hybrid systems. Illustrative examples concerning unpredictable\nsolutions of differential equations are provided.", + "category": "nlin_CD" + }, + { + "text": "Genesis of d'Alembert's paradox and analytical elaboration of the drag\n problem: We show that the issue of the drag exerted by an incompressible fluid on a\nbody in uniform motion has played a major role in the early development of\nfluid dynamics. In 1745 Euler came close, technically, to proving the vanishing\nof the drag for a body of arbitrary shape; for this he exploited and\nsignificantly extended existing ideas on decomposing the flow into thin\nfillets; he did not however have a correct picture of the global structure of\nthe flow around a body. Borda in 1766 showed that the principle of live forces\nimplied the vanishing of the drag and should thus be inapplicable to the\nproblem. After having at first refused the possibility of a vanishing drag,\nd'Alembert in 1768 established the paradox, but only for bodies with a\nhead-tail symmetry. A full understanding of the paradox, as due to the neglect\nof viscous forces, had to wait until the work of Saint-Venant in 1846.", + "category": "nlin_CD" + }, + { + "text": "Rich-club network topology to minimize synchronization cost due to phase\n difference among frequency-synchronized oscillators: Functions of some networks, such as power grids and large-scale brain\nnetworks, rely on not only frequency synchronization, but also phase\nsynchronization. Nevertheless, even after the oscillators reach to\nfrequency-synchronized status, phase difference among oscillators often shows\nnon-zero constant values. Such phase difference potentially results in\ninefficient transfer of power or information among oscillators, and avoid\nproper and efficient functioning of the network. In the present study, we newly\ndefine synchronization cost by the phase difference among the\nfrequency-synchronized oscillators, and investigate the optimal network\nstructure with the minimum synchronization cost through rewiring-based\noptimization. By using the Kuramoto model, we demonstrate that the cost is\nminimized in a network topology with rich-club organization, which comprises\nthe densely-connected center nodes and peripheral nodes connecting with the\ncenter module. We also show that the network topology is characterized by its\nbimodal degree distribution, which is quantified by Wolfson's polarization\nindex. Furthermore, we provide analytical interpretation on why the rich-club\nnetwork topology is related to the small amount of synchronization cost.", + "category": "nlin_CD" + }, + { + "text": "Quantum Graphs: A simple model for Chaotic Scattering: We connect quantum graphs with infinite leads, and turn them to scattering\nsystems. We show that they display all the features which characterize quantum\nscattering systems with an underlying classical chaotic dynamics: typical\npoles, delay time and conductance distributions, Ericson fluctuations, and when\nconsidered statistically, the ensemble of scattering matrices reproduce quite\nwell the predictions of appropriately defined Random Matrix ensembles. The\nunderlying classical dynamics can be defined, and it provides important\nparameters which are needed for the quantum theory. In particular, we derive\nexact expressions for the scattering matrix, and an exact trace formula for the\ndensity of resonances, in terms of classical orbits, analogous to the\nsemiclassical theory of chaotic scattering. We use this in order to investigate\nthe origin of the connection between Random Matrix Theory and the underlying\nclassical chaotic dynamics. Being an exact theory, and due to its relative\nsimplicity, it offers new insights into this problem which is at the fore-front\nof the research in chaotic scattering and related fields.", + "category": "nlin_CD" + }, + { + "text": "Dissipative structures in a nonlinear dynamo: This paper considers magnetic field generation by a fluid flow in a system\nreferred to as the Archontis dynamo: a steady nonlinear magnetohydrodynamic\n(MHD) state is driven by a prescribed body force. The field and flow become\nalmost equal and dissipation is concentrated in cigar-like structures centred\non straight-line separatrices. Numerical scaling laws for energy and\ndissipation are given that extend previous calculations to smaller\ndiffusivities. The symmetries of the dynamo are set out, together with their\nimplications for the structure of field and flow along the separatrices. The\nscaling of the cigar-like dissipative regions, as the square root of the\ndiffusivities, is explained by approximations near the separatrices. Rigorous\nresults on the existence and smoothness of solutions to the steady, forced MHD\nequations are given.", + "category": "nlin_CD" + }, + { + "text": "How winding is the coast of Britain ? Conformal invariance of rocky\n shorelines: We show that rocky shorelines with fractal dimension 4/3 are conformally\ninvariant curves by measuring the statistics of their winding angles from\nglobal high-resolution data. Such coastlines are thus statistically equivalent\nto the outer boundary of the random walk and of percolation clusters. A simple\nmodel of coastal erosion gives an explanation for these results. Conformal\ninvariance allows also to predict the highly intermittent spatial distribution\nof the flux of pollutant diffusing ashore.", + "category": "nlin_CD" + }, + { + "text": "Computing the multifractal spectrum from time series: An algorithmic\n approach: We show that the existing methods for computing the f(\\alpha) spectrum from a\ntime series can be improved by using a new algorithmic scheme. The scheme\nrelies on the basic idea that the smooth convex profile of a typical f(\\alpha)\nspectrum can be fitted with an analytic function involving a set of four\nindependent parameters. While the standard existing schemes [16, 18] generally\ncompute only an incomplete f(\\alpha) spectrum (usually the top portion), we\nshow that this can be overcome by an algorithmic approach which is automated to\ncompute the Dq and f(\\alpha) spectrum from a time series for any embedding\ndimension. The scheme is first tested with the logistic attractor with known\nf(\\alpha) curve and subsequently applied to higher dimensional cases. We also\nshow that the scheme can be effectively adapted for analysing practcal time\nseries involving noise, with examples from two widely different real world\nsystems. Moreover, some preliminary results indicating that the set of four\nindependant parameters may be used as diagnostic measures is also included.", + "category": "nlin_CD" + }, + { + "text": "Detection of Generalized Synchronization using Echo State Networks: Generalized synchronization between coupled dynamical systems is a phenomenon\nof relevance in applications that range from secure communications to\nphysiological modelling. Here we test the capabilities of reservoir computing\nand, in particular, echo state networks for the detection of generalized\nsynchronization. A nonlinear dynamical system consisting of two coupled\nR\\\"ossler chaotic attractors is used to generate temporal series consisting of\ntime-locked generalized synchronized sequences interleaved by unsynchronized\nones. Correctly tuned, echo state networks are able to efficiently discriminate\nbetween unsynchronized and synchronized sequences. Compared to other\nstate-of-the-art techniques of synchronization detection, the online\ncapabilities of the proposed ESN based methodology make it a promising choice\nfor real-time applications aiming to monitor dynamical synchronization changes\nin continuous signals.", + "category": "nlin_CD" + }, + { + "text": "Strange nonchaotic stars: The unprecedented light curves of the Kepler space telescope document how the\nbrightness of some stars pulsates at primary and secondary frequencies whose\nratios are near the golden mean, the most irrational number. A nonlinear\ndynamical system driven by an irrational ratio of frequencies generically\nexhibits a strange but nonchaotic attractor. For Kepler's \"golden\" stars, we\npresent evidence of the first observation of strange nonchaotic dynamics in\nnature outside the laboratory. This discovery could aid the classification and\ndetailed modeling of variable stars.", + "category": "nlin_CD" + }, + { + "text": "Cryptanalysis of a one round chaos-based Substitution Permutation\n Network: The interleaving of chaos and cryptography has been the aim of a large set of\nworks since the beginning of the nineties. Many encryption proposals have been\nintroduced to improve conventional cryptography. However, many proposals\npossess serious problems according to the basic requirements for the secure\nexchange of information. In this paper we highlight some of the main problems\nof chaotic cryptography by means of the analysis of a very recent chaotic\ncryptosystem based on a one round Substitution Permutation Network. More\nspecifically, we show that it is not possible to avoid the security problems of\nthat encryption architecture just by including a chaotic system as core of the\nderived encryption system.", + "category": "nlin_CD" + }, + { + "text": "Pseudo-random number generator based on asymptotic deterministic\n randomness: An approach to generate the pseudorandom-bit sequence from the asymptotic\ndeterministic randomness system is proposed in this Letter. We study the\ncharacteristic of multi-value correspondence of the asymptotic deterministic\nrandomness constructed by the piecewise linear map and the noninvertible\nnonlinearity transform, and then give the discretized systems in the finite\ndigitized state space. The statistic characteristics of the asymptotic\ndeterministic randomness are investigated numerically, such as stationary\nprobability density function and random-like behavior. Furthermore, we analyze\nthe dynamics of the symbolic sequence. Both theoretical and experimental\nresults show that the symbolic sequence of the asymptotic deterministic\nrandomness possesses very good cryptographic properties, which improve the\nsecurity of chaos based PRBGs and increase the resistance against entropy\nattacks and symbolic dynamics attacks.", + "category": "nlin_CD" + }, + { + "text": "Interactions destroy dynamical localization with strong and weak chaos: Bose-Einstein condensates loaded into kicked optical lattices can be treated\nas quantum kicked rotor systems. Noninteracting rotors show dynamical\nlocalization in momentum space. The experimentally tunable condensate\ninteraction is included in a qualitative Gross-Pitaevskii type model based on\ntwo-body interactions. We observe strong and weak chaos regimes of wave packet\nspreading in momentum space. In the intermediate strong chaos regime the\ncondensate energy grows as $t^{1/2}$. In the asymptotic weak chaos case the\ngrowth crosses over into a $t^{1/3}$ law. The results do not depend on the\ndetails of the kicking.", + "category": "nlin_CD" + }, + { + "text": "Networked control systems: a perspective from chaos: In this paper, a nonlinear system aiming at reducing the signal transmission\nrate in a networked control system is constructed by adding nonlinear\nconstraints to a linear feedback control system. Its stability is investigated\nin detail. It turns out that this nonlinear system exhibits very interesting\ndynamical behaviors: in addition to local stability, its trajectories may\nconverge to a non-origin equilibrium or be periodic or just be oscillatory.\nFurthermore it exhibits sensitive dependence on initial conditions --- a sign\nof chaos. Complicated bifurcation phenomena are exhibited by this system. After\nthat, control of the chaotic system is discussed. All these are studied under\nscalar cases in detail. Some difficulties involved in the study of this type of\nsystems are analyzed. Finally an example is employed to reveal the\neffectiveness of the scheme in the framework of networked control systems.", + "category": "nlin_CD" + }, + { + "text": "Dynamic synchronization of a time-evolving optical network of chaotic\n oscillators: We present and experimentally demonstrate a technique for achieving and\nmaintaining a global state of identical synchrony of an arbitrary network of\nchaotic oscillators even when the coupling strengths are unknown and\ntime-varying. At each node an adaptive synchronization algorithm dynamically\nestimates the current strength of the net coupling signal to that node. We\nexperimentally demonstrate this scheme in a network of three bidirectionally\ncoupled chaotic optoelectronic feedback loops and we present numerical\nsimulations showing its application in larger networks. The stability of the\nsynchronous state for arbitrary coupling topologies is analyzed via a master\nstability function approach.", + "category": "nlin_CD" + }, + { + "text": "Map representation of the time-delayed system in the presence of Delay\n Time Modulation: an Application to the stability analysis: We introduce the map representation of a time-delayed system in the presence\nof delay time modulation. Based on this representation, we find the method by\nwhich to analyze the stability of that kind of a system. We apply this method\nto a coupled chaotic system and discuss the results in comparison to the system\nwith a fixed delay time.", + "category": "nlin_CD" + }, + { + "text": "Simulation studies on the design of optimum PID controllers to suppress\n chaotic oscillations in a family of Lorenz-like multi-wing attractors: Multi-wing chaotic attractors are highly complex nonlinear dynamical systems\nwith higher number of index-2 equilibrium points. Due to the presence of\nseveral equilibrium points, randomness and hence the complexity of the state\ntime series for these multi-wing chaotic systems is much higher than that of\nthe conventional double-wing chaotic attractors. A real-coded Genetic Algorithm\n(GA) based global optimization framework has been adopted in this paper as a\ncommon template for designing optimum Proportional-Integral-Derivative (PID)\ncontrollers in order to control the state trajectories of four different\nmulti-wing chaotic systems among the Lorenz family viz. Lu system, Chen system,\nRucklidge (or Shimizu Morioka) system and Sprott-1 system. Robustness of the\ncontrol scheme for different initial conditions of the multi-wing chaotic\nsystems has also been shown.", + "category": "nlin_CD" + }, + { + "text": "Riddling: Chimera's dilemma: We investigate the basin of attraction properties and its boundaries for\nchimera states in a circulant network of H\\'enon maps. Chimera states, for\nwhich coherent and incoherent domains coexist, emerge as a consequence of the\ncoexistence of basin of attractions for each state. It is known that the\ncoexisting basins of attraction lead to a hysteretic behaviour in the diagrams\nfor the density of incoherent and coherent states as a function of a varying\nparameter. Consequently, the distribution of chimera states can remain\ninvariant by a parameter change, as well as it can suffer subtle changes when\none of the basin ceases to exist. A similar phenomenon is observed when\nperturbations are applied in the initial conditions. By means of the\nuncertainty exponent, we characterise the basin boundaries between the coherent\nand chimera states, and between the incoherent and chimera states, and uncover\nfractal and riddled boundaries, respectively. This way, we show that the\ndensity of chimera states can be not only moderately sensitive but also highly\nsensitive to initial conditions. This chimera's dilemma is a consequence of the\nfractal and riddled nature of the basins boundaries.", + "category": "nlin_CD" + }, + { + "text": "Effective stochastic model for chaos in the Fermi-Pasta-Ulam-Tsingou\n chain: Understanding the interplay between different wave excitations, such as\nphonons and localized solitons, is crucial for developing coarse-grained\ndescriptions of many-body, near-integrable systems. We treat the\nFermi-Pasta-Ulam-Tsingou (FPUT) non-linear chain and show numerically that at\nshort timescales, relevant to the largest Lyapunov exponent, it can be modeled\nas a random perturbation of its integrable approximation -- the Toda chain. At\nlow energies, the separation between two trajectories that start at close\nproximity is dictated by the interaction between few soliton modes and an\nintrinsic, apparent bath representing a background of many radiative modes. It\nis sufficient to consider only one randomly perturbed Toda soliton-like mode to\nexplain the power-law profiles reported in previous works, describing how the\nLyapunov exponent of large FPUT chains decreases with the energy density of the\nsystem.", + "category": "nlin_CD" + }, + { + "text": "Using Synchronization for Prediction of High-Dimensional Chaotic\n Dynamics: We experimentally observe the nonlinear dynamics of an optoelectronic\ntime-delayed feedback loop designed for chaotic communication using commercial\nfiber optic links, and we simulate the system using delay differential\nequations. We show that synchronization of a numerical model to experimental\nmeasurements provides a new way to assimilate data and forecast the future of\nthis time-delayed high-dimensional system. For this system, which has a\nfeedback time delay of 22 ns, we show that one can predict the time series for\nup to several delay periods, when the dynamics is about 15 dimensional.", + "category": "nlin_CD" + }, + { + "text": "Non-ergodicity and localization of invariant measure for two colliding\n masses: We show evidence, based on extensive and carefully performed numerical\nexperiments, that the system of two elastic hard-point masses in one-dimension\nis not ergodic for a generic mass ratio and consequently does not follow the\nprinciple of energy equipartition. This system is equivalent to a right\ntriangular billiard. Remarkably, following the time-dependent probability\ndistribution in a suitably chosen velocity direction space, we find evidence of\nexponential localization of invariant measure. For non-generic mass ratios\nwhich correspond to billiard angles which are rational, or weak irrational\nmultiples of pi, the system is ergodic, in consistence with existing rigorous\nresults.", + "category": "nlin_CD" + }, + { + "text": "Radial disk heating by more than one spiral density wave: We consider a differentially rotating, 2D stellar disk perturbed by two\nsteady state spiral density waves moving at different patterns speeds. Our\ninvestigation is based on direct numerical integration of initially circular\ntest-particle orbits. We examine a range of spiral strengths and spiral speeds\nand show that stars in this time dependent gravitational field can be heated\n(their random motions increased). This is particularly noticeable in the\nsimultaneous propagation of a 2-armed spiral density wave near the corotation\nresonance (CR), and a weak 4-armed one near the inner and outer 4:1 Lindblad\nresonances. In simulations with 2 spiral waves moving at different pattern\nspeeds we find: (1) the variance of the radial velocity, sigma_R^2, exceeds the\nsum of the variances measured from simulations with each individual pattern;\n(2) sigma_R^2 can grow with time throughout the entire simulation; (3)\nsigma_R^2 is increased over a wider range of radii compared to that seen with\none spiral pattern; (4) particles diffuse radially in real space whereas they\ndon't when only one spiral density wave is present. Near the CR with the\nstronger, 2-armed pattern, test particles are observed to migrate radially.\nThese effects take place at or near resonances of both spirals so we interpret\nthem as the result of stochastic motions. This provides a possible new\nmechanism for increasing the stellar velocity dispersion in galactic disks. If\nmultiple spiral patterns are present in the Galaxy we predict that there should\nbe large variations in the stellar velocity dispersion as a function of radius.", + "category": "nlin_CD" + }, + { + "text": "Singular continuous spectra in a pseudo-integrable billiard: The pseudo-integrable barrier billiard invented by Hannay and McCraw [J.\nPhys. A 23, 887 (1990)] -- rectangular billiard with line-segment barrier\nplaced on a symmetry axis -- is generalized. It is proven that the flow on\ninvariant surfaces of genus two exhibits a singular continuous spectral\ncomponent.", + "category": "nlin_CD" + }, + { + "text": "Localization Properties of Covariant Lyapunov Vectors: The Lyapunov exponent spectrum and covariant Lyapunov vectors are studied for\na quasi-one-dimensional system of hard disks as a function of density and\nsystem size. We characterize the system using the angle distributions between\ncovariant vectors and the localization properties of both Gram-Schmidt and\ncovariant vectors. At low density there is a {\\it kinetic regime} that has\nsimple scaling properties for the Lyapunov exponents and the average\nlocalization for part of the spectrum. This regime shows strong localization in\na proportion of the first Gram-Schmidt and covariant vectors and this can be\nunderstood as highly localized configurations dominating the vector. The\ndistribution of angles between neighbouring covariant vectors has\ncharacteristic shapes depending upon the difference in vector number, which\nvary over the continuous region of the spectrum. At dense gas or liquid like\ndensities the behaviour of the covariant vectors are quite different. The\npossibility of tangencies between different components of the unstable manifold\nand between the stable and unstable manifolds is explored but it appears that\nexact tangencies do not occur for a generic chaotic trajectory.", + "category": "nlin_CD" + }, + { + "text": "Transport in Hamiltonian systems with slowly changing phase space\n structure: Transport in Hamiltonian systems with weak chaotic perturbations has been\nmuch studied in the past. In this paper, we introduce a new class of problems:\ntransport in Hamiltonian systems with slowly changing phase space structure\nthat are not order one perturbations of a given Hamiltonian. This class of\nproblems is very important for many applications, for instance in celestial\nmechanics. As an example, we study a class of one-dimensional Hamiltonians that\ndepend explicitly on time and on stochastic external parameters. The variations\nof the external parameters are responsible for a distortion of the phase space\nstructures: chaotic, weakly chaotic and regular sets change with time. We show\nthat theoretical predictions of transport rates can be made in the limit where\nthe variations of the stochastic parameters are very slow compared to the\nHamiltonian dynamics. Exact asymptotic results can be obtained in the classical\ncase where the Hamiltonian dynamics is integrable for fixed values of the\nparameters. For the more interesting chaotic Hamiltonian dynamics case, we show\nthat two mechanisms contribute to the transport. For some range of the\nparameter variations, one mechanism -called transport by migration together\nwith the mixing regions - is dominant. We are then able to model transport in\nphase space by a Markov model, the local diffusion model, and to give\nreasonably good transport estimates.", + "category": "nlin_CD" + }, + { + "text": "Poincare recurrences from the perspective of transient chaos: We obtain a description of the Poincar\\'e recurrences of chaotic systems in\nterms of the ergodic theory of transient chaos. It is based on the equivalence\nbetween the recurrence time distribution and an escape time distribution\nobtained by leaking the system and taking a special initial ensemble. This\nensemble is atypical in terms of the natural measure of the leaked system, the\nconditionally invariant measure. Accordingly, for general initial ensembles,\nthe average recurrence and escape times are different. However, we show that\nthe decay rate of these distributions is always the same. Our results remain\nvalid for Hamiltonian systems with mixed phase space and validate a split of\nthe chaotic saddle in hyperbolic and non-hyperbolic components.", + "category": "nlin_CD" + }, + { + "text": "Quasiclassical Born-Oppenheimer approximations: We discuss several problems in quasiclassical physics for which approximate\nsolutions were recently obtained by a new method, and which can also be solved\nby novel versions of the Born-Oppenheimer approximation. These cases include\nthe so-called bouncing ball modes, low angular momentum states in perturbed\ncircular billiards, resonant states in perturbed rectangular billiards, and\nwhispering gallery modes. Some rare, special eigenstates, concentrated close to\nthe edge or along a diagonal of a nearly rectangular billiard are found. This\nkind of state has apparently previously escaped notice.", + "category": "nlin_CD" + }, + { + "text": "The effect of noise on a hyperbolic strange attractor in the system of\n two coupled van der Pol oscillators: We study the effect of noise for a physically realizable flow system with a\nhyperbolic chaotic attractor of the Smale - Williams type in the Poincare\ncross-section [S.P. Kuznetsov, Phys. Rev. Lett. 95, 2005, 144101]. It is shown\nnumerically that slightly varying the initial conditions on the attractor one\ncan obtain a uniform approximation of a noisy orbit by the trajectory of the\nsystem without noise, that is called as the \"shadowing\" trajectory. We propose\nan algorithm for locating the shadowing trajectories in the system under\nconsideration. Using this algorithm, we show that the mean distance between a\nnoisy orbit and the approximating one does not depend essentially on the length\nof the time interval of observation, but only on the noise intensity. This\ndependance is nearly linear in a wide interval of the intensities of noise. It\nis found out that for weak noise the Lyapunov exponents do not depend\nnoticeably on the noise intensity. However, in the case of a strong noise the\nlargest Lyapunov exponent decreases and even becomes negative indicating the\nsuppression of chaos by the external noise.", + "category": "nlin_CD" + }, + { + "text": "Entropic comparison of Landau-Zener and Demkov interactions in the phase\n space of a quadrupole billiard: We investigate two types of avoided crossings in a chaotic billiard within\nthe framework of information theory. The Shannon entropy in the phase space for\nthe Landau--Zener interaction increases as the center of the avoided crossing\nis approached. Meanwhile, that for the Demkov interaction decreases as the\ncenter of avoided crossing is passed by with an increase in the deformation\nparameter. This feature can provide a new indicator for scar formation. In\naddition, it is found that the Fisher information of the Landau--Zener\ninteraction is significantly larger than that of the Demkov interaction.", + "category": "nlin_CD" + }, + { + "text": "Behavior of Dynamical Systems in the Regime of Transient Chaos: The transient chaos regime in a two-dimensional system with discrete time\n(Eno map) is considered. It is demonstrated that a time series corresponding to\nthis regime differs from a chaotic series constructed for close values of the\ncontrol parameters by the presence of \"nonregular\" regions, the number of which\nincreases with the critical parameter. A possible mechanism of this effect is\ndiscussed.", + "category": "nlin_CD" + }, + { + "text": "The largest Lyapunov exponent as a tool for detecting relative changes\n in the particle positions: Dynamics of the driven Frenkel-Kontorova model with asymmetric deformable\nsubstrate potential is examined by analyzing response function, the largest\nLyapunov exponent and Poincar\\'{e} sections for two neighboring particles. The\nobtained results show that the largest Lyapunov exponent, besides being used\nfor investigating integral quantities, can be used for detecting microchanges\nin chain configuration of both damped Frenkel-Kontorova model with inertial\nterm and its strictly overdamped limit. Slight changes in relative positions of\nthe particles are registered through jumps of the largest Lyapunov exponent in\nthe pinning regime. The occurrence of such jumps is highly dependent on type of\ncommensurate structure and deformation of substrate potential. The obtained\nresults also show that the minimal force required to initiate collective motion\nof the chain is not dependent on the number of Lyapunov exponent jumps in the\npinning regime. These jumps are also registered in the sliding regime, where\nthey are a consequence of a more complex structure of largest Lyapunov exponent\non the step.", + "category": "nlin_CD" + }, + { + "text": "Zeta Function Zeros, Powers of Primes, and Quantum Chaos: We present a numerical study of Riemann's formula for the oscillating part of\nthe density of the primes and their powers. The formula is comprised of an\ninfinite series of oscillatory terms, one for each zero of the zeta function on\nthe critical line and was derived by Riemann in his paper on primes assuming\nthe Riemann hypothesis. We show that high resolution spectral lines can be\ngenerated by the truncated series at all powers of primes and demonstrate\nexplicitly that the relative line intensities are correct. We then derive a\nGaussian sum rule for Riemann's formula. This is used to analyze the numerical\nconvergence of the truncated series. The connections to quantum chaos and\nsemiclassical physics are discussed.", + "category": "nlin_CD" + }, + { + "text": "Determinism, Complexity, and Predictability in Computer Performance: Computers are deterministic dynamical systems (CHAOS 19:033124, 2009). Among\nother things, that implies that one should be able to use deterministic\nforecast rules to predict their behavior. That statement is sometimes-but not\nalways-true. The memory and processor loads of some simple programs are easy to\npredict, for example, but those of more-complex programs like compilers are\nnot. The goal of this paper is to determine why that is the case. We conjecture\nthat, in practice, complexity can effectively overwhelm the predictive power of\ndeterministic forecast models. To explore that, we build models of a number of\nperformance traces from different programs running on different Intel-based\ncomputers. We then calculate the permutation entropy-a temporal entropy metric\nthat uses ordinal analysis-of those traces and correlate those values against\nthe prediction success", + "category": "nlin_CD" + }, + { + "text": "Chimera states in networks of nonlocally coupled Hindmarsh-Rose neuron\n models: We have identified the occurrence of chimera states for various coupling\nschemes in networks of two-dimensional and three-dimensional Hindmarsh-Rose\noscillators, which represent realistic models of neuronal ensembles. This\nresult, together with recent studies on multiple chimera states in nonlocally\ncoupled FitzHugh-Nagumo oscillators, provide strong evidence that the\nphenomenon of chimeras may indeed be relevant in neuroscience applications.\nMoreover, our work verifies the existence of chimera states in coupled bistable\nelements, whereas to date chimeras were known to arise in models possessing a\nsingle stable limit cycle. Finally, we have identified an interesting class of\nmixed oscillatory states, in which desynchronized neurons are uniformly\ninterspersed among the remaining ones that are either stationary or oscillate\nin synchronized motion.", + "category": "nlin_CD" + }, + { + "text": "Bichromatically driven double well: parametric perspective of the\n strong-field control landscape reveals the influence of chaotic states: The aim of this work is to understand the influence of chaotic states in\ncontrol problems involving strong fields. Towards this end, we numerically\nconstruct and study the strong field control landscape of a bichromatically\ndriven double well. A novel measure based on correlating the overlap\nintensities between Floquet states and an initial phase space coherent state\nwith the parametric motion of the quasienergies is used to construct and\ninterpret the landscape features. \"Walls\" of no control, robust under\nvariations of the relative phase between the fields, are seen on the control\nlandscape and associated with multilevel interactions involving chaotic Floquet\nstates.", + "category": "nlin_CD" + }, + { + "text": "Chaotic flow and efficient mixing in a micro-channel with a polymer\n solution: Microscopic flows are almost universally linear, laminar and stationary\nbecause Reynolds number, $Re$, is usually very small. That impedes mixing in\nmicro-fluidic devices, which sometimes limits their performance. Here we show\nthat truly chaotic flow can be generated in a smooth micro-channel of a uniform\nwidth at arbitrarily low $Re$, if a small amount of flexible polymers is added\nto the working liquid. The chaotic flow regime is characterized by randomly\nfluctuating three-dimensional velocity field and significant growth of the flow\nresistance. Although the size of the polymer molecules extended in the flow may\nbecome comparable with the micro-channel width, the flow behavior is fully\ncompatible with that in a table-top channel in the regime of elastic\nturbulence. The chaotic flow leads to quite efficient mixing, which is almost\ndiffusion independent. For macromolecules, mixing time in this microscopic flow\ncan be three to four orders of magnitude shorter than due to molecular\ndiffusion.", + "category": "nlin_CD" + }, + { + "text": "Measures of Anisotropy and the Universal Properties of Turbulence: Local isotropy, or the statistical isotropy of small scales, is one of the\nbasic assumptions underlying Kolmogorov's theory of universality of small-scale\nturbulent motion. While, until the mid-seventies or so, local isotropy was\naccepted as a plausible approximation at high enough Reynolds numbers, various\nempirical observations that have accumulated since then suggest that local\nisotropy may not obtain at any Reynolds number. These notes examine in some\ndetail the isotropic and anisotropic contributions to structure functions by\nconsidering their SO(3) decomposition. Viewed in terms of the relative\nimportance of the isotropic part to the anisotropic parts of structure\nfunctions, the basic conclusion is that the isotropic part dominates the small\nscales at least up to order 6. This follows from the fact that, at least up to\nthat order, there exists a hierarchy of increasingly larger power-law\nexponents, corresponding to increasingly higher-order anisotropic sectors of\nthe SO(3) decomposition. The numerical values of the exponents deduced from\nexperiment suggest that the anisotropic parts in each order roll off less\nsharply than previously thought by dimensional considerations, but they do so\nnevertheless.", + "category": "nlin_CD" + }, + { + "text": "A non-autonomous flow system with Plykin type attractor: A non-autonomous flow system is introduced with an attractor of Plykin type\nthat may serve as a base for elaboration of real systems and devices\ndemonstrating the structurally stable chaotic dynamics. The starting point is a\nmap on a two-dimensional sphere, consisting of four stages of continuous\ngeometrically evident transformations. The computations indicate that in a\ncertain parameter range the map has a uniformly hyperbolic attractor. It may be\nrepresented on a plane by means of a stereographic projection. Accounting\nstructural stability, a modification of the model is undertaken to obtain a set\nof two non-autonomous differential equations of the first order with smooth\ncoefficients. As follows from computations, it has the Plykin type attractor in\nthe Poincar\\'{e} cross-section.", + "category": "nlin_CD" + }, + { + "text": "Dependence of heat transport on the strength and shear rate of\n prescribed circulating flows: We study numerically the dependence of heat transport on the maximum velocity\nand shear rate of physical circulating flows, which are prescribed to have the\nkey characteristics of the large-scale mean flow observed in turbulent\nconvection. When the side-boundary thermal layer is thinner than the viscous\nboundary layer, the Nusselt number (Nu), which measures the heat transport,\nscales with the normalized shear rate to an exponent 1/3. On the other hand,\nwhen the side-boundary thermal layer is thicker, the dependence of Nu on the\nPeclet number, which measures the maximum velocity, or the normalized shear\nrate when the viscous boundary layer thickness is fixed, is generally not a\npower law. Scaling behavior is obtained only in an asymptotic regime. The\nrelevance of our results to the problem of heat transport in turbulent\nconvection is also discussed.", + "category": "nlin_CD" + }, + { + "text": "On near integrability of some impact systems: A class of Hamiltonian impact systems exhibiting smooth near integrable\nbehavior is presented. The underlying unperturbed model investigated is an\nintegrable, separable, 2 degrees of freedom mechanical impact system with\neffectively bounded energy level sets and a single straight wall which\npreserves the separable structure. Singularities in the system appear either as\ntrajectories with tangent impacts or as singularities in the underlying\nHamiltonian structure (e.g. separatrices). It is shown that away from these\nsingularities, a small perturbation from the integrable structure results in\nsmooth near integrable behavior. Such a perturbation may occur from a small\ndeformation or tilt of the wall which breaks the separability upon impact, the\naddition of a small regular perturbation to the system, or the combination of\nboth. In some simple cases explicit formulae to the leading order term in the\nnear integrable return map are derived. Near integrability is also shown to\npersist when the hard billiard boundary is replaced by a singular, smooth,\nsteep potential, thus extending the near-integrability results beyond the scope\nof regular perturbations. These systems constitute an additional class of\nexamples of near integrable impact systems, beyond the traditional one\ndimensional oscillating billiards, nearly elliptic billiards, and the\nnear-integrable behavior near the boundary of convex smooth billiards with or\nwithout magnetic field.", + "category": "nlin_CD" + }, + { + "text": "Delay sober up drunkers: Control of diffusion in random walkers: Time delay in general leads to instability in some systems, while a specific\nfeedback with delay can control fluctuated motion in nonlinear deterministic\nsystems to a stable state. In this paper, we consider a non-stationary\nstochastic process, i.e., a random walk and observe its diffusion phenomenon\nwith time delayed feedback. Surprisingly, the diffusion coefficient decreases\nwith increasing the delay time. We analytically illustrate this suppression of\ndiffusion by using stochastic delay differential equations and justify the\nfeasibility of this suppression by applying the time-delay feedback to a\nmolecular dynamics model.", + "category": "nlin_CD" + }, + { + "text": "Modeling Kelvin Wave Cascades in Superfluid Helium: We study two different types of simplified models for Kelvin wave turbulence\non quantized vortex lines in superfluids near zero temperature. Our first model\nis obtained from a truncated expansion of the Local Induction Approximation\n(Truncated-LIA) and it is shown to possess the same scalings and the essential\nbehaviour as the full Biot-Savart model, being much simpler than the latter\nand, therefore, more amenable to theoretical and numerical investigations. The\nTruncated-LIA model supports six-wave interactions and dual cascades, which are\nclearly demonstrated via the direct numerical simulation of this model in the\npresent paper. In particular, our simulations confirm presence of the weak\nturbulence regime and the theoretically predicted spectra for the direct energy\ncascade and the inverse wave action cascade. The second type of model we study,\nthe Differential Approximation Model (DAM), takes a further drastic\nsimplification by assuming locality of interactions in $k$-space via a\ndifferential closure that preserves the main scalings of the Kelvin wave\ndynamics. DAMs are even more amenable to study and they form a useful tool by\nproviding simple analytical solutions in the cases when extra physical effects\nare present, e.g. forcing by reconnections, friction dissipation and phonon\nradiation. We study these models numerically and test their theoretical\npredictions, in particular the formation of the stationary spectra, and the\ncloseness of the numerics for the higher-order DAM to the analytical\npredictions for the lower-order DAM .", + "category": "nlin_CD" + }, + { + "text": "Sensitivity of long periodic orbits of chaotic systems: The properties of long, numerically-determined periodic orbits of two\nlow-dimensional chaotic systems, the Lorenz equations and the\nKuramoto-Sivashinsky system in a minimal-domain configuration, are examined.\nThe primary question is to establish whether the sensitivity of period averaged\nquantities with respect to parameter perturbations computed over long orbits\ncan be used as a sufficiently good proxy for the response of the chaotic state\nto finite-amplitude parameter perturbations. To address this question, an\ninventory of thousands of orbits at least two orders of magnitude longer than\nthe shortest admissible cycles is constructed. The expectation of period\naverages, Floquet exponents and sensitivities over such set is then obtained.\nIt is shown that all these quantities converge to a limiting value as the orbit\nperiod is increased. However, while period averages and Floquet exponents\nappear to converge to analogous quantities computed from chaotic trajectories,\nthe limiting value of the sensitivity is not necessarily consistent with the\nresponse of the chaotic state, similar to observations made with other\nshadowing algorithms.", + "category": "nlin_CD" + }, + { + "text": "Energy and potential enstrophy flux constraints in the two-layer\n quasi-geostrophic model: We investigate an inequality constraining the energy and potential enstrophy\nflux in the two-layer quasi-geostrophic model. This flux inequality is\nunconditionally satisfied for the case of two-dimensional Navier-Stokes\nturbulence. However, it is not obvious that it remains valid under the\nmulti-layer quasi-geostrophic model. The physical significance of this\ninequality is that it decides whether any given model can reproduce the\nNastrom-Gage spectrum of the atmosphere, at least in terms of the total energy\nspectrum. We derive the general form of the energy and potential enstrophy\ndissipation rate spectra for a generalized multi-layer model. We then\nspecialize these results for the case of the two-layer quasi-geostrophic model\nunder dissipation configurations in which the dissipation terms for each layer\nare dependent only on the streamfunction or potential vorticity of that layer.\nWe derive sufficient conditions for satisfying the flux inequality and discuss\nthe possibility of violating it under different conditions.", + "category": "nlin_CD" + }, + { + "text": "Ergodicity, mixing and recurrence in the three rotor problem: In the classical three rotor problem, three equal point masses move on a\ncircle subject to attractive cosine potentials of strength g. In the center of\nmass frame, energy E is the only known conserved quantity. In earlier work\n[Krishnaswami and Senapati, arXiv:1810.01317, Oct. 2018, arXiv:1811.05807, Nov.\n2018], an order-chaos-order transition was discovered in this system along with\na band of global chaos for 5.33g < E < 5.6g. Here, we provide numerical\nevidence for ergodicity and mixing in this band. The distributions of relative\nangles and angular momenta along generic trajectories are shown to approach the\ncorresponding distributions over constant energy hypersurfaces (weighted by the\nLiouville measure) as a power-law in time. Moreover, trajectories emanating\nfrom a small volume are shown to become uniformly distributed over constant\nenergy hypersurfaces, indicating that the dynamics is mixing. Outside this\nband, ergodicity and mixing fail, though the distributions of angular momenta\nover constant energy hypersurfaces show interesting phase transitions from\nWignerian to bimodal with increasing energy. Finally, in the band of global\nchaos, the distribution of recurrence times to finite size cells is found to\nfollow an exponential law with the mean recurrence time satisfying a scaling\nlaw involving an exponent consistent with global chaos and ergodicity.", + "category": "nlin_CD" + }, + { + "text": "Intermittency and Universality in Fully-Developed Inviscid and\n Weakly-Compressible Turbulent Flows: We performed high resolution numerical simulations of homogenous and\nisotropic compressible turbulence, with an average 3D Mach number close to 0.3.\nWe study the statistical properties of intermittency for velocity, density and\nentropy. For the velocity field, which is the primary quantity that can be\ncompared to the isotropic incompressible case, we find no statistical\ndifferences in its behavior in the inertial range due either to the slight\ncompressibility or to the different dissipative mechanism. For the density\nfield, we find evidence of ``front-like'' structures, although no shocks are\nproduced by the simulation.", + "category": "nlin_CD" + }, + { + "text": "Dynamical tunneling and control: This article summarizes the recent work on the influence of dynamical\ntunneling on the control of quantum systems. Specifically, two examples are\ndiscussed. In the first, it is shown that the bichromatic control of tunneling\nin a driven double well system is hampered by the phenomenon of chaos-assisted\ntunneling. The bichromatic control landscape exhibits several regions\nindicating lack of control with every such region involving chaos-assisted\ntunneling. The second example illustrates the failure of controlling the\ndissociation dynamics of a driven Morse oscillator due to the phenomenon of\nresonance-assisted tunneling. In particular, attempts to control the\ndissociation dynamics by rebuilding local phase space barriers are foiled due\nto resonance-assisted tunneling.", + "category": "nlin_CD" + }, + { + "text": "Generalized Amplitude Truncation of Gaussian 1/f^alpha noise: We study a kind of filtering, an amplitude truncation with upper and lower\ntruncation levels x_max and x_min. This is a generalization of the simple\ntransformation y(t)=sgn[x(t)], for which a rigorous result was obtained\nrecently. So far numerical experiments have shown that a power law spectrum\n1/f^alpha seems to be transformed again into a power law spectrum 1/f^beta\nunder rather general condition for the truncation levels. We examine the above\nnumerical results analytically. When 1 T(zz) > T(yy).", + "category": "nlin_CD" + }, + { + "text": "Twenty-five years of multifractals in fully developed turbulence: a\n tribute to Giovanni Paladin: The paper {\\it On the multifractal nature of fully developed turbulence and\nchaotic systems}, by R. Benzi {\\it et al.} published in this journal in 1984\n(vol {\\bf 17}, page 3521) has been a starting point of many investigations on\nthe different faces of selfsimilarity and intermittency in turbulent phenomena.\n Since then, the multifractal model has become a useful tool for the study of\nsmall scale turbulence, in particular for detailed predictions of different\nEulerian and Lagrangian statistical properties. In the occasion of the 50-th\nbirthday of our unforgettable friend and colleague Giovanni Paladin\n(1958-1996), we review here the basic concepts and some applications of the\nmultifractal model for turbulence.", + "category": "nlin_CD" + }, + { + "text": "Elastic turbulence in a polymer solution flow: Turbulence is one of the most fascinating phenomena in nature and one of the\nbiggest challenges for modern physics. It is common knowledge that a flow of a\nsimple, Newtonian fluid is likely to be turbulent, when velocity is high,\nviscosity is low and size of the tank is large\\cite{Landau,Tritt}. Solutions of\nflexible long-chain polymers are known as visco-elastic fluids\\cite{bird}. In\nour experiments we show, that flow of a polymer solution with large enough\nelasticity can become quite irregular even at low velocity, high viscosity and\nin a small tank. The fluid motion is excited in a broad range of spatial and\ntemporal scales. The flow resistance increases by a factor of about twenty. So,\nwhile the Reynolds number, $\\boldmath{Re}$, may be arbitrary low, the observed\nflow has all main features of developed turbulence, and can be compared to\nturbulent flow in a pipe at $\\bf {Re\\simeq 10^5}$\\cite{Landau,Tritt}. This {\\it\nelastic turbulence} is accompanied by significant stretching of the polymer\nmolecules, and the resulting increase of the elastic stresses can reach two\norders of magnitude.", + "category": "nlin_CD" + }, + { + "text": "A note on dissipation in helical turbulence: In helical turbulence a linear cascade of helicity accompanying the energy\ncascade has been suggested. Since energy and helicity have different\ndimensionality we suggest the existence of a characteristic inner scale,\n$\\xi=k_H^{-1}$, for helicity dissipation in a regime of hydrodynamic fully\ndeveloped turbulence and estimate it on dimensional grounds. This scale is\nalways larger than the Kolmogorov scale, $\\eta=k_E^{-1}$, and their ratio $\\eta\n/ \\xi $ vanishes in the high Reynolds number limit, so the flow will always be\nhelicity free in the small scales.", + "category": "nlin_CD" + }, + { + "text": "Universal behaviour of a wave chaos based electromagnetic reverberation\n chamber: In this article, we present a numerical investigation of three-dimensional\nelectromagnetic Sinai-like cavities. We computed around 600 eigenmodes for two\ndifferent geometries: a parallelepipedic cavity with one half- sphere on one\nwall and a parallelepipedic cavity with one half-sphere and two spherical caps\non three adjacent walls. We show that the statistical requirements of a well\noperating reverberation chamber are better satisfied in the more complex\ngeometry without a mechanical mode-stirrer/tuner. This is to the fact that our\nproposed cavities exhibit spatial and spectral statistical behaviours very\nclose to those predicted by random matrix theory. More specifically, we show\nthat in the range of frequency corresponding to the first few hundred modes,\nthe suppression of non-generic modes (regarding their spatial statistics) can\nbe achieved by reducing drastically the amount of parallel walls. Finally, we\ncompare the influence of losses on the statistical complex response of the\nfield inside a parallelepipedic and a chaotic cavity. We demonstrate that, in a\nchaotic cavity without any stirring process, the low frequency limit of a well\noperating reverberation chamber can be significantly reduced under the usual\nvalues obtained in mode-stirred reverberation chambers.", + "category": "nlin_CD" + }, + { + "text": "Resonance states of the three-disk scattering system: For the paradigmatic three-disk scattering system, we confirm a recent\nconjecture for open chaotic systems, which claims that resonance states are\ncomposed of two factors. In particular, we demonstrate that one factor is given\nby universal exponentially distributed intensity fluctuations. The other\nfactor, supposed to be a classical density depending on the lifetime of the\nresonance state, is found to be very well described by a classical\nconstruction. Furthermore, ray-segment scars, recently observed in dielectric\ncavities, dominate every resonance state at small wavelengths also in the\nthree-disk scattering system. We introduce a new numerical method for computing\nresonances, which allows for going much further into the semiclassical limit.\nAs a consequence we are able to confirm the fractal Weyl law over a\ncorrespondingly large range.", + "category": "nlin_CD" + }, + { + "text": "The Decay of Passive Scalars Under the Action of Single Scale Smooth\n Velocity Fields in Bounded 2D Domains : From non self similar pdf's to self\n similar eigenmodes: We examine the decay of passive scalars with small, but non zero, diffusivity\nin bounded 2D domains. The velocity fields responsible for advection are smooth\n(i.e., they have bounded gradients) and of a single large scale. Moreover, the\nscale of the velocity field is taken to be similar to the size of the entire\ndomain. The importance of the initial scale of variation of the scalar field\nwith respect to that of the velocity field is strongly emphasized. If these\nscales are comparable and the velocity field is time periodic, we see the\nformation of a periodic scalar eigenmode. The eigenmode is numerically realized\nby means of a deterministic 2D map on a lattice. Analytical justification for\nthe eigenmode is available from theorems in the dynamo literature. Weakening\nthe notion of an eigenmode to mean statistical stationarity, we provide\nnumerical evidence that the eigenmode solution also holds for aperiodic flows\n(represented by random maps). Turning to the evolution of an initially small\nscale scalar field, we demonstrate the transition from an evolving (i.e., {\\it\nnon} self similar) pdf to a stationary (self similar) pdf as the scale of\nvariation of the scalar field progresses from being small to being comparable\nto that of the velocity field (and of the domain). Furthermore, the {\\it non}\nself similar regime itself consists of two stages. Both the stages are examined\nand the coupling between diffusion and the distribution of the Finite Time\nLyapunov Exponents is shown to be responsible for the pdf evolution.", + "category": "nlin_CD" + }, + { + "text": "Intermittency and Synchronisation in Gumowski-Mira Maps: The Gumowski-Mira map is a 2-dimensional recurrence relation that provide a\nlarge variety of phase space plots resembling fractal patterns of nature. We\ninvestigate the nature of the dynamical states that produce these patterns and\nfind that they correspond to Type I intermittency near periodic cycles. By\ncoupling two GM maps, such patterns can be wiped out to give synchronised\nperiodic states of lower order.The efficiency of the coupling scheme is\nestablished by analysing the error function dynamics.", + "category": "nlin_CD" + }, + { + "text": "A self-synchronizing stream cipher based on chaotic coupled maps: A revised self-synchronizing stream cipher based on chaotic coupled maps is\nproposed. This system adds input and output functions aim to strengthen its\nsecurity. The system performs basic floating-point analytical computation on\nreal numbers, incorporating auxiliarily with algebraic operations on integer\nnumbers.", + "category": "nlin_CD" + }, + { + "text": "The overlapping of nonlinear resonances and problem of quantum chaos: The motion of a nonlinearly oscilating partical under the influence of a\nperiodic sequence of short impulses is investigated. We analyze the Schrodinger\nequation for the universal Hamiltonian. The idea about the emerging of quantum\nchaos due to the adiabatic motion along the curves of Mathieu characteristics\nat multiple passages through the points of branching is advanced", + "category": "nlin_CD" + }, + { + "text": "The Dynamical Matching Mechanism in Phase Space for Caldera-Type\n Potential Energy Surfaces: Dynamical matching occurs in a variety of important organic chemical\nreactions. It is observed to be a result of a potential energy surface (PES)\nhaving specific geometric features. In particular, a region of relative\nflatness where entrance and exit to this region is controlled by index-one\nsaddles. Examples of potential energy surfaces having these features are the\nso-called caldera potential energy surfaces. We develop a predictive level of\nunderstanding of the phenomenon of dynamical matching in a caldera potential\nenergy surface. We show that the phase space structure that governs dynamical\nmatching is a particular type of heteroclinic trajectory which gives rise to\ntrapping of trajectories in the central region of the caldera PES. When the\nheteroclinic trajectory is broken, as a result of parameter variations, then\ndynamical matching occurs.", + "category": "nlin_CD" + }, + { + "text": "The oscillating two-cluster chimera state in non-locally coupled phase\n oscillators: We investigate an array of identical phase oscillators non-locally coupled\nwithout time delay, and find that chimera state with two coherent clusters\nexists which is only reported in delay-coupled systems previously. Moreover, we\nfind that the chimera state is not stationary for any finite number of\noscillators. The existence of the two-cluster chimera state and its\ntime-dependent behaviors for finite number of oscillators are confirmed by the\ntheoretical analysis based on the self-consistency treatment and the\nOtt-Antonsen ansatz.", + "category": "nlin_CD" + }, + { + "text": "Experimental test of a trace formula for two-dimensional dielectric\n resonators: Resonance spectra of two-dimensional dielectric microwave resonators of\ncircular and square shapes have been measured. The deduced length spectra of\nperiodic orbits were analyzed and a trace formula for dielectric resonators\nrecently proposed by Bogomolny et al. [Phys. Rev. E 78, 056202 (2008)] was\ntested. The observed deviations between the experimental length spectra and the\npredictions of the trace formula are attributed to a large number of missing\nresonances in the measured spectra. We show that by taking into account the\nsystematics of observed and missing resonances the experimental length spectra\nare fully understood. In particular, a connection between the most long-lived\nresonances and certain periodic orbits is established experimentally.", + "category": "nlin_CD" + }, + { + "text": "Dynamical Analysis of a Networked Control System: A new network data transmission strategy was proposed in Zhang \\& Chen [2005]\n(arXiv:1405.2404), where the resulting nonlinear system was analyzed and the\neffectiveness of the transmission strategy was demonstrated via simulations. In\nthis paper, we further generalize the results of Zhang \\& Chen [2005] in the\nfollowing ways: 1) Construct first-return maps of the nonlinear systems\nformulated in Zhang \\& Chen [2005] and derive several existence conditions of\nperiodic orbits and study their properties. 2) Formulate the new system as a\nhybrid system, which will ease the succeeding analysis. 3) Prove that this type\nof hybrid systems is not structurally stable based on phase transition which\ncan be applied to higher-dimensional cases effortlessly. 4) Simulate a\nhigher-dimensional model with emphasis on their rich dynamics. 5) Study a class\nof continuous-time hybrid systems as the counterparts of the discrete-time\nsystems discussed above. 6) Propose new controller design methods based on this\nnetwork data transmission strategy to improve the performance of each\nindividual system and the whole network. We hope that this research and the\nproblems posed here will rouse interests of researchers in such fields as\ncontrol, dynamical systems and numerical analysis.", + "category": "nlin_CD" + }, + { + "text": "Influence of weak anisotropy on scaling regimes in a model of advected\n vector field: Influence of weak uniaxial small-scale anisotropy on the stability of\ninertial-range scaling regimes in a model of a passive transverse vector field\nadvected by an incompressible turbulent flow is investigated by means of the\nfield theoretic renormalization group. Weak anisotropy means that parameters\nwhich describe anisotropy are chosen to be close to zero, therefore in all\nexpressions it is enough to leave only linear terms in anisotropy parameters.\nTurbulent fluctuations of the velocity field are taken to have the Gaussian\nstatistics with zero mean and defined noise with finite correlations in time.\nIt is shown that stability of the inertial-range scaling regimes in the\nthree-dimensional case is not destroyed by anisotropy but the corresponding\nstability of the two-dimensional system can be destroyed even by the presence\nof weak anisotropy. A borderline dimension $d_c$ below which the stability of\nthe scaling regime is not present is calculated as a function of anisotropy\nparameters.", + "category": "nlin_CD" + }, + { + "text": "Peeling Bifurcations of Toroidal Chaotic Attractors: Chaotic attractors with toroidal topology (van der Pol attractor) have\ncounterparts with symmetry that exhibit unfamiliar phenomena. We investigate\ndouble covers of toroidal attractors, discuss changes in their morphology under\ncorrelated peeling bifurcations, describe their topological structures and the\nchanges undergone as a symmetry axis crosses the original attractor, and\nindicate how the symbol name of a trajectory in the original lifts to one in\nthe cover. Covering orbits are described using a powerful synthesis of kneading\ntheory with refinements of the circle map. These methods are applied to a\nsimple version of the van der Pol oscillator.", + "category": "nlin_CD" + }, + { + "text": "Low-frequency regime transitions and predictability of regimes in a\n barotropic model: Predictability of flow is examined in a barotropic vorticity model that\nadmits low frequency regime transitions between zonal and dipolar states. Such\ntransitions in the model were first studied by Bouchet and Simonnet (2009) and\nare reminiscent of regime change phenomena in the weather and climate systems\nwherein extreme and abrupt qualitative changes occur, seemingly randomly, after\nlong periods of apparent stability. Mechanisms underlying regime transitions in\nthe model are not well understood yet. From the point of view of atmospheric\nand oceanic dynamics, a novel aspect of the model is the lack of any source of\nbackground gradient of potential-vorticity such as topography or planetary\ngradient of rotation rate (e.g., as in Charney & DeVore '79).\n We consider perturbations that are embedded onto the system's chaotic\nattractor under the full nonlinear dynamics as bred vectors---nonlinear\ngeneralizations of the leading (backward) Lyapunov vector. We find that\nensemble predictions that use bred vector perturbations are more robust in\nterms of error-spread relationship than those that use Lyapunov vector\nperturbations. In particular, when bred vector perturbations are used in\nconjunction with a simple data assimilation scheme (nudging to truth), we find\nthat at least some of the evolved perturbations align to identify\nlow-dimensional subspaces associated with regions of large forecast error in\nthe control (unperturbed, data-assimilating) run; this happens less often in\nensemble predictions that use Lyapunov vector perturbations. Nevertheless, in\nthe inertial regime we consider, we find that (a) the system is more\npredictable when it is in the zonal regime, and that (b) the horizon of\npredictability is far too short compared to characteristic time scales\nassociated with processes that lead to regime transitions, thus precluding the\npossibility of predicting such transitions.", + "category": "nlin_CD" + }, + { + "text": "Extended Harmonic Map Equations and the Chaotic Soliton Solutions: In this paper, the theory of harmonic maps is extended. The soliton or\ntraveling wave solutions of Euler's equations of the extended harmonic maps are\nstudied. In certain cases, the chaotic behaviors of these partial equations can\nbe found for the particular case of the metrics and the potential functions of\nthe extended harmonic equations.", + "category": "nlin_CD" + }, + { + "text": "Standard map-like models for single and multiple walkers in an annular\n cavity: Recent experiments on walking droplets in an annular cavity showed the\nexistence of complex dynamics including chaotically changing velocity. This\narticle presents models, influenced by the kicked rotator/standard map, for\nboth single and multiple droplets. The models are shown to achieve both\nqualitative and quantitative agreement with the experiments, and makes\npredictions about heretofore unobserved behavior. Using dynamical systems\ntechniques and bifurcation theory, the single droplet model is analyzed to\nprove dynamics suggested by the numerical simulations.", + "category": "nlin_CD" + }, + { + "text": "Non-integrability of restricted double pendula: We consider two special types of double pendula, with the motion of masses\nrestricted to various surfaces. In order to get quick insight into the dynamics\nof the considered systems the Poincar\\'e cross sections as well as bifurcation\ndiagrams have been used. The numerical computations show that both models are\nchaotic which suggest that they are not integrable. We give an analytic proof\nof this fact checking the properties of the differential Galois group of the\nsystem's variational equations along a particular non-equilibrium solution.", + "category": "nlin_CD" + }, + { + "text": "Perturbation-Free Prediction of Resonance-Assisted Tunneling in Mixed\n Regular--Chaotic Systems: For generic Hamiltonian systems we derive predictions for dynamical tunneling\nfrom regular to chaotic phase-space regions. In contrast to previous\napproaches, we account for the resonance-assisted enhancement of\nregular-to-chaotic tunneling in a non-perturbative way. This provides the\nfoundation for future semiclassical complex-path evaluations of\nresonance-assisted regular-to-chaotic tunneling. Our approach is based on a new\nclass of integrable approximations which mimic the regular phase-space region\nand its dominant nonlinear resonance chain in a mixed regular--chaotic system.\nWe illustrate the method for the standard map.", + "category": "nlin_CD" + }, + { + "text": "Echoes in classical dynamical systems: Echoes arise when external manipulations to a system induce a reversal of its\ntime evolution that leads to a more or less perfect recovery of the initial\nstate. We discuss the accuracy with which a cloud of trajectories returns to\nthe initial state in classical dynamical systems that are exposed to additive\nnoise and small differences in the equations of motion for forward and backward\nevolution. The cases of integrable and chaotic motion and small or large noise\nare studied in some detail and many different dynamical laws are identified.\nExperimental tests in 2-d flows that show chaotic advection are proposed.", + "category": "nlin_CD" + }, + { + "text": "Rattling and freezing in a 1-D transport model: We consider a heat conduction model introduced in \\cite{Collet-Eckmann 2009}.\nThis is an open system in which particles exchange momentum with a row of\n(fixed) scatterers. We assume simplified bath conditions throughout, and give a\nqualitative description of the dynamics extrapolating from the case of a single\nparticle for which we have a fairly clear understanding. The main phenomenon\ndiscussed is {\\it freezing}, or the slowing down of particles with time. As\nparticle number is conserved, this means fewer collisions per unit time, and\nless contact with the baths; in other words, the conductor becomes less\neffective. Careful numerical documentation of freezing is provided, and a\ntheoretical explanation is proposed. Freezing being an extremely slow process,\nhowever, the system behaves as though it is in a steady state for long\ndurations. Quantities such as energy and fluxes are studied, and are found to\nhave curious relationships with particle density.", + "category": "nlin_CD" + }, + { + "text": "On nonlinear fractional maps: Nonlinear maps with power-law memory: This article is a short review of the recent results on properties of\nnonlinear fractional maps which are maps with power- or asymptotically\npower-law memory. These maps demonstrate the new type of attractors - cascade\nof bifurcations type trajectories, power-law convergence/divergence of\ntrajectories, period doubling bifurcations with changes in the memory\nparameter, intersection of trajectories, and overlapping of attractors. In the\nlimit of small time steps these maps converge to nonlinear fractional\ndifferential equations.", + "category": "nlin_CD" + }, + { + "text": "Intermittent Peel Front Dynamics and the Crackling Noise in an Adhesive\n Tape: We report a comprehensive investigation of a model for peeling of an adhesive\ntape along with a nonlinear time series analysis of experimental acoustic\nemission signals in an effort to understand the origin of intermittent peeling\nof an adhesive tape and its connection to acoustic emission. The model\nrepresents the acoustic energy dissipated in terms of Rayleigh dissipation\nfunctional that depends on the local strain rate. We show that the nature of\nthe peel front exhibits rich spatiotemporal patterns ranging from smooth,\nrugged and stuck-peeled configurations that depend on three parameters, namely,\nthe ratio of inertial time scale of the tape mass to that of the roller, the\ndissipation coefficient and the pull velocity. The stuck-peeled configurations\nare reminiscent of fibrillar peel front patterns observed in experiments. We\nshow that while the intermittent peeling is controlled by the peel force\nfunction, the model acoustic energy dissipated depends on the nature of the\npeel front and its dynamical evolution. Even though the acoustic energy is a\nfully dynamical quantity, it can be quite noisy for a certain set of parameter\nvalues suggesting the deterministic origin of acoustic emission in experiments.\nTo verify this suggestion, we have carried out a dynamical analysis of\nexperimental acoustic emission time series for a wide range of traction\nvelocities. Our analysis shows an unambiguous presence of chaotic dynamics\nwithin a subinterval of pull speeds within the intermittent regime. Time series\nanalysis of the model acoustic energy signals is also found to be chaotic\nwithin a subinterval of pull speeds.", + "category": "nlin_CD" + }, + { + "text": "Explosive synchronization transition in a ring of coupled oscillators: Explosive synchronization(ES), as one kind of abrupt dynamical transition in\nnonlinearly coupled systems, is currently a subject of great interests. Given a\nspecial frequency distribution, a mixed ES is observed in a ring of coupled\nphase oscillators which transit from partial synchronization to ES with the\nincrement of coupling strength. The coupling weight is found to control the\nsize of the hysteresis region where asynchronous and synchronized states\ncoexist. Theoretical analysis reveals that the transition varies from the mixed\nES, to the ES and then to a continuous one with increasing coupling weight. Our\nresults are helpful to extend the understanding of the ES in homogenous\nnetworks.", + "category": "nlin_CD" + }, + { + "text": "Multi-locality and fusion rules on the generalized structure functions\n in two-dimensional and three-dimensional Navier-Stokes turbulence: Using the fusion rules hypothesis for three-dimensional and two-dimensional\nNavier-Stokes turbulence, we generalize a previous non-perturbative locality\nproof to multiple applications of the nonlinear interactions operator on\ngeneralized structure functions of velocity differences. We shall call this\ngeneralization of non-perturbative locality to multiple applications of the\nnonlinear interactions operator \"multilocality\". The resulting cross-terms pose\na new challenge requiring a new argument and the introduction of a new fusion\nrule that takes advantage of rotational symmetry. Our main result is that the\nfusion rules hypothesis implies both locality and multilocality in both the IR\nand UV limits for the downscale energy cascade of three-dimensional\nNavier-Stokes turbulence and the downscale enstrophy cascade and inverse energy\ncascade of two-dimensional Navier-Stokes turbulence. We stress that these\nclaims relate to non-perturbative locality of generalized structure functions\non all orders, and not the term by term perturbative locality of diagrammatic\ntheories or closure models that involve only two-point correlation and response\nfunctions.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of Three Non-co-rotating Vortices in Bose-Einstein Condensates: In this work we use standard Hamiltonian-system techniques in order to study\nthe dynamics of three vortices with alternating charges in a confined\nBose-Einstein condensate. In addition to being motivated by recent experiments,\nthis system offers a natural vehicle for the exploration of the transition of\nthe vortex dynamics from ordered to progressively chaotic behavior. In\nparticular, it possesses two integrals of motion, the {\\it energy} (which is\nexpressed through the Hamiltonian $H$) and the {\\it angular momentum} $L$ of\nthe system. By using the integral of the angular momentum, we reduce the system\nto a two degree-of-freedom one with $L$ as a parameter and reveal the topology\nof the phase space through the method of Poincar\\'e surfaces of section.\n We categorize the various motions that appear in the different regions of the\nsections and we study the major bifurcations that occur to the families of\nperiodic motions of the system. Finally, we correspond the orbits on the\nsurfaces of section to the real space motion of the vortices in the plane.", + "category": "nlin_CD" + }, + { + "text": "Chaotic mechanism description by an elementary mixer for the template of\n an attractor: Templates can be used to describe the topological properties of chaotic\nattractors. For attractors bounded by genus one torus, these templates are\ndescribed by a linking matrix. For a given attractor, it has been shown that\nthe template depends on the Poincar\\'e section chosen to performed the\nanalysis. The purpose of this article is to present an algorithm that gives the\nelementary mixer of a template in order to have a unique way to describe a\nchaotic mechanism. This chaotic mechanism is described with a linking matrix\nand we also provide a method to generate and classify all the possible chaotic\nmechanisms made of two to five strips.", + "category": "nlin_CD" + }, + { + "text": "Chimera-Like Coexistence of Synchronized Oscillation and Death in an\n Ecological Network: We report a novel spatiotemporal state, namely the chimera-like incongruous\ncoexistence of {\\it synchronized oscillation} and {\\it stable steady state}\n(CSOD) in a realistic ecological network of nonlocally coupled oscillators.\nUnlike the {\\it chimera} and {\\it chimera death} state, in the CSOD state\nidentical oscillators are self-organized into two coexisting spatially\nseparated domains: In one domain neighboring oscillators show synchronized\noscillation and in another domain the neighboring oscillators randomly populate\neither a synchronized oscillating state or a stable steady state (we call it a\ndeath state). We show that the interplay of nonlocality and coupling strength\nresults in two routes to the CSOD state: One is from a coexisting mixed state\nof amplitude chimera and death, and another one is from a globally synchronized\nstate. We further explore the importance of this study in ecology that gives a\nnew insight into the relationship between spatial synchrony and global\nextinction of species.", + "category": "nlin_CD" + }, + { + "text": "Chaos on a High-Dimensional Torus: Transition from quasiperiodicity with many frequencies (i.e., a\nhigh-dimensional torus) to chaos is studied by using $N$-dimensional globally\ncoupled circle maps. First, the existence of $N$-dimensional tori with $N\\geq\n2$ is confirmed while they become exponentially rare with $N$. Besides, chaos\nexists even when the map is invertible, and such chaos has more null Lyapunov\nexponents as $N$ increases. This unusual form of \"chaos on a torus,\" termed\ntoric chaos, exhibits delocalization and slow dynamics of the first Lyapunov\nvector. Fractalization of tori at the transition to chaos is also suggested.\nThe relevance of toric chaos to neural dynamics and turbulence is discussed in\nrelation to chaotic itinerancy.", + "category": "nlin_CD" + }, + { + "text": "Relaxation of finite perturbations: Beyond the Fluctuation-Response\n relation: We study the response of dynamical systems to finite amplitude perturbation.\nA generalized Fluctuation-Response relation is derived, which links the average\nrelaxation toward equilibrium to the invariant measure of the system and points\nout the relevance of the amplitude of the initial perturbation. Numerical\ncomputations on systems with many characteristic times show the relevance of\nthe above relation in realistic cases.", + "category": "nlin_CD" + }, + { + "text": "Assessing the direction of climate interactions by means of complex\n networks and information theoretic tools: An estimate of the net direction of climate interactions in different\ngeographical regions is made by constructing a directed climate network from a\nregular latitude-longitude grid of nodes, using a directionality index (DI)\nbased on conditional mutual information. Two datasets of surface air\ntemperature anomalies - one monthly-averaged and another daily-averaged - are\nanalyzed and compared. The network links are interpreted in terms of known\natmospheric tropical and extratropical variability patterns. Specific and\nrelevant geographical regions are selected, the net direction of propagation of\nthe atmospheric patterns is analyzed and the direction of the inferred links is\nvalidated by recovering some well-known climate variability structures. These\npatterns are found to be acting at various time-scales, such as atmospheric\nwaves in the extra-tropics or longer range events in the tropics. This analysis\ndemonstrates the capability of the DI measure to infer the net direction of\nclimate interactions and may contribute to improve the present understanding of\nclimate phenomena and climate predictability. The work presented here also\nstands out as an application of advanced tools to the analysis of empirical,\nreal-world data.", + "category": "nlin_CD" + }, + { + "text": "Experimental observation of a complex periodic window: The existence of a special periodic window in the two-dimensional parameter\nspace of an experimental Chua's circuit is reported. One of the main reasons\nthat makes such a window special is that the observation of one implies that\nother similar periodic windows must exist for other parameter values. However,\nsuch a window has never been experimentally observed, since its size in\nparameter space decreases exponentially with the period of the periodic\nattractor. This property imposes clear limitations for its experimental\ndetection.", + "category": "nlin_CD" + }, + { + "text": "Estimating short-time period to break different types of chaotic\n modulation based secure communications: In recent years, chaotic attractors have been extensively used in the design\nof secure communication systems. One of the preferred ways of transmitting the\ninformation signal is binary chaotic modulation, in which a binary message\nmodulates a parameter of the chaotic generator. This paper presents a method of\nattack based on estimating the short-time period of the ciphertext generated\nfrom the modulated chaotic attractor. By calculating and then filtering the\nshort-time period of the transmitted signal it is possible to obtain the binary\ninformation signal with great accuracy without any knowledge of the parameters\nof the underlying chaotic system. This method is successfully applied to\nvarious secure communication systems proposed in the literature based on\ndifferent chaotic attractors.", + "category": "nlin_CD" + }, + { + "text": "Variational principles in the analysis of traffic flows. (Why it is\n worth to go against the flow.): By means of a novel variational approach and using dual maps techniques and\ngeneral ideas of dynamical system theory we derive exact results about several\nmodels of transport flows, for which we also obtain a complete description of\ntheir limit (in time) behavior in the space of configurations. Using these\nresults we study the motion of a speedy passive particle (tracer) moving\nalong/against the flow of slow particles and demonstrate that the latter case\nmight be more efficient.", + "category": "nlin_CD" + }, + { + "text": "Rough basin boundaries in high dimension: Can we classify them\n experimentally?: We show that a known condition for having rough basin boundaries in bistable\n2D maps holds for high-dimensional bistable systems that possess a unique\nnonattracting chaotic set embedded in their basin boundaries. The condition for\nroughness is that the cross-boundary Lyapunov exponent $\\lambda_x$ {\\bfac on\nthe nonattracting set} is not the maximal one. Furthermore, we provide a\nformula for the generally noninteger co-dimension of the rough basin boundary,\nwhich can be viewed as a generalization of the Kantz-Grassberger formula. This\nco-dimension that can be at most unity can be thought of as a partial\nco-dimension, and, so, it can be matched with a Lyapunov exponent. We show\n{\\bfac in 2D noninvertible- and 3D invertible minimal models,} that, formally,\nit cannot be matched with $\\lambda_x$. Rather, the partial dimension\n$D_0^{(x)}$ that $\\lambda_x$ is associated with in the case of rough boundaries\nis trivially unity. Further results hint that the latter holds also in higher\ndimensions. This is a peculiar feature of rough fractals. Yet, $D_0^{(x)}$\ncannot be measured via the uncertainty exponent along a line that traverses the\nboundary. Indeed, one cannot determine whether the boundary is a rough or a\nfilamentary fractal by measuring fractal dimensions. Instead, one needs to\nmeasure both the maximal and cross-boundary Lyapunov exponents numerically or\nexperimentally.", + "category": "nlin_CD" + }, + { + "text": "Learning to imitate stochastic time series in a compositional way by\n chaos: This study shows that a mixture of RNN experts model can acquire the ability\nto generate sequences combining multiple primitive patterns by means of\nself-organizing chaos. By training of the model, each expert learns a primitive\nsequence pattern, and a gating network learns to imitate stochastic switching\nof the multiple primitives via a chaotic dynamics, utilizing a sensitive\ndependence on initial conditions. As a demonstration, we present a numerical\nsimulation in which the model learns Markov chain switching among some\nLissajous curves by a chaotic dynamics. Our analysis shows that by using a\nsufficient amount of training data, balanced with the network memory capacity,\nit is possible to satisfy the conditions for embedding the target stochastic\nsequences into a chaotic dynamical system. It is also shown that reconstruction\nof a stochastic time series by a chaotic model can be stabilized by adding a\nnegligible amount of noise to the dynamics of the model.", + "category": "nlin_CD" + }, + { + "text": "Design of time delayed chaotic circuit with threshold controller: A novel time delayed chaotic oscillator exhibiting mono- and double scroll\ncomplex chaotic attractors is designed. This circuit consists of only a few\noperational amplifiers and diodes and employs a threshold controller for\nflexibility. It efficiently implements a piecewise linear function. The control\nof piecewise linear function facilitates controlling the shape of the\nattractors. This is demonstrated by constructing the phase portraits of the\nattractors through numerical simulations and hardware experiments. Based on\nthese studies, we find that this circuit can produce multi-scroll chaotic\nattractors by just introducing more number of threshold values.", + "category": "nlin_CD" + }, + { + "text": "Fronts in passive scalar turbulence: The evolution of scalar fields transported by turbulent flow is characterized\nby the presence of fronts, which rule the small-scale statistics of scalar\nfluctuations. With the aid of numerical simulations, it is shown that: isotropy\nis not recovered, in the classical sense, at small scales; scaling exponents\nare universal with respect to the scalar injection mechanisms; high-order\nexponents saturate to a constant value; non-mature fronts dominate the\nstatistics of intense fluctuations. Results on the statistics inside the\nplateaux, where fluctuations are weak, are also presented. Finally, we analyze\nthe statistics of scalar dissipation and scalar fluxes.", + "category": "nlin_CD" + }, + { + "text": "Zero tension Kardar-Parisi-Zhang equation in (d+1)- Dimensions: The joint probability distribution function (PDF) of the height and its\ngradients is derived for a zero tension $d+1$-dimensional Kardar-Parisi-Zhang\n(KPZ) equation. It is proved that the height`s PDF of zero tension KPZ equation\nshows lack of positivity after a finite time $t_{c}$. The properties of zero\ntension KPZ equation and its differences with the case that it possess an\ninfinitesimal surface tension is discussed. Also potential relation between the\ntime scale $t_{c}$ and the singularity time scale $t_{c, \\nu \\to 0}$ of the KPZ\nequation with an infinitesimal surface tension is investigated.", + "category": "nlin_CD" + }, + { + "text": "Nests and Chains of Hofstadter Butterflies: The \\lq Hofstadter butterfly', a plot of the spectrum of an electron in a\ntwo-dimensional periodic potential with a uniform magnetic field, contains\nsubsets which resemble small, distorted images of the entire plot. We show how\nthe sizes of these sub-images are determined, and calculate scaling factors\ndescribing their self-similar nesting, revealing an un-expected simplicity in\nthe fractal structure of the spectrum. We also characterise semi-infinite\nchains of sub-images, showing one end of the chain is a result of gap closure,\nand the other end is at an accumulation point.", + "category": "nlin_CD" + }, + { + "text": "Generalized synchronization in mutually coupled oscillators and complex\n networks: We introduce a novel concept of generalized synchronization, able to\nencompass the setting of collective synchronized behavior for mutually coupled\nsystems and networking systems featuring complex topologies in their\nconnections. The onset of the synchronous regime is confirmed by the dependence\nof the system's Lyapunov exponents on the coupling parameter. The presence of a\ngeneralized synchronization regime is verified by means of the nearest neighbor\nmethod.", + "category": "nlin_CD" + }, + { + "text": "Transition from amplitude to oscillation death under mean-field\n diffusive coupling: We study the transition from amplitude death (AD) to oscillation death (OD)\nstate in limit-cycle oscillators coupled through mean-field diffusion. We show\nthat this coupling scheme can induce an important transition from AD to OD even\nin {\\it identical} limit cycle oscillators. We identify a parameter region\nwhere OD and a novel {\\it nontrivial} AD (NT-AD) state coexist. This NT-AD\nstate is unique in comparison with AD owing to the fact that it is created by a\nsubcritical pitchfork bifurcation, and parameter mismatch does not support but\ndestroy this state. We extend our study to a network of mean-field coupled\noscillators to show that the transition scenario preserves and the oscillators\nform a two cluster state.", + "category": "nlin_CD" + }, + { + "text": "Evanescent wave approach to diffractive phenomena in convex billiards\n with corners: What we are going to call in this paper \"diffractive phenomena\" in billiards\nis far from being deeply understood. These are sorts of singularities that, for\nexample, some kind of corners introduce in the energy eigenfunctions. In this\npaper we use the well-known scaling quantization procedure to study them. We\nshow how the scaling method can be applied to convex billiards with corners,\ntaking into account the strong diffraction at them and the techniques needed to\nsolve their Helmholtz equation. As an example we study a classically\npseudointegrable billiard, the truncated triangle. Then we focus our attention\non the spectral behavior. A numerical study of the statistical properties of\nhigh-lying energy levels is carried out. It is found that all computed\nstatistical quantities are roughly described by the so-called semi-Poisson\nstatistics, but it is not clear whether the semi-Poisson statistics is the\ncorrect one in the semiclassical limit.", + "category": "nlin_CD" + }, + { + "text": "Symbolic dynamics techniques for complex systems: Application to share\n price dynamics: The symbolic dynamics technique is well-known for low-dimensional dynamical\nsystems and chaotic maps, and lies at the roots of the thermodynamic formalism\nof dynamical systems. Here we show that this technique can also be successfully\napplied to time series generated by complex systems of much higher\ndimensionality. Our main example is the investigation of share price returns in\na coarse-grained way. A nontrivial spectrum of Renyi entropies is found. We\nstudy how the spectrum depends on the time scale of returns, the sector of\nstocks considered, as well as the number of symbols used for the symbolic\ndescription. Overall our analysis confirms that in the symbol space transition\nprobabilities of observed share price returns depend on the entire history of\nprevious symbols, thus emphasizing the need for a modelling based on\nnon-Markovian stochastic processes. Our method allows for quantitative\ncomparisons of entirely different complex systems, for example the statistics\nof symbol sequences generated by share price returns using 4 symbols can be\ncompared with that of genomic sequences.", + "category": "nlin_CD" + }, + { + "text": "One-Particle and Few-Particle Billiards: We study the dynamics of one-particle and few-particle billiard systems in\ncontainers of various shapes. In few-particle systems, the particles collide\nelastically both against the boundary and against each other. In the\none-particle case, we investigate the formation and destruction of resonance\nislands in (generalized) mushroom billiards, which are a recently discovered\nclass of Hamiltonian systems with mixed regular-chaotic dynamics. In the\nfew-particle case, we compare the dynamics in container geometries whose\ncounterpart one-particle billiards are integrable, chaotic, and mixed. One of\nour findings is that two-, three-, and four-particle billiards confined to\ncontainers with integrable one-particle counterparts inherit some integrals of\nmotion and exhibit a regular partition of phase space into ergodic components\nof positive measure. Therefore, the shape of a container matters not only for\nnoninteracting particles but also for interacting particles.", + "category": "nlin_CD" + }, + { + "text": "Ehrenfest times for classically chaotic systems: We describe the quantum mechanical spreading of a Gaussian wave packet by\nmeans of the semiclassical WKB approximation of Berry and Balazs. We find that\nthe time scale $\\tau$ on which this approximation breaks down in a chaotic\nsystem is larger than the Ehrenfest times considered previously. In one\ndimension $\\tau=\\fr{7}{6}\\lambda^{-1}\\ln(A/\\hbar)$, with $\\lambda$ the Lyapunov\nexponent and $A$ a typical classical action.", + "category": "nlin_CD" + }, + { + "text": "Mapping Model of Chaotic Phase Synchronization: A coupled map model for the chaotic phase synchronization and its\ndesynchronization phenomenon is proposed. The model is constructed by\nintegrating the coupled kicked oscillator system, kicking strength depending on\nthe complex state variables. It is shown that the proposed model clearly\nexhibits the chaotic phase synchronization phenomenon. Furthermore, we\nnumerically prove that in the region where the phase synchronization is weakly\nbroken, the anomalous scaling of the phase difference rotation number is\nobserved. This proves that the present model belongs to the same universality\nclass found by Pikovsky et al.. Furthermore, the phase diffusion coefficient in\nthe de-synchronization state is analyzed.", + "category": "nlin_CD" + }, + { + "text": "Boundary-induced instabilities in coupled oscillators: A novel class of nonequilibrium phase-transitions at zero temperature is\nfound in chains of nonlinear oscillators.For two paradigmatic systems, the\nHamiltonian XY model and the discrete nonlinear Schr\\\"odinger equation, we find\nthat the application of boundary forces induces two synchronized phases,\nseparated by a non-trivial interfacial region where the kinetic temperature is\nfinite. Dynamics in such supercritical state displays anomalous chaotic\nproperties whereby some observables are non-extensive and transport is\nsuperdiffusive. At finite temperatures, the transition is smoothed, but the\ntemperature profile is still non-monotonous.", + "category": "nlin_CD" + }, + { + "text": "Python for Education: Computational Methods for Nonlinear Systems: We describe a novel, interdisciplinary, computational methods course that\nuses Python and associated numerical and visualization libraries to enable\nstudents to implement simulations for a number of different course modules.\nProblems in complex networks, biomechanics, pattern formation, and gene\nregulation are highlighted to illustrate the breadth and flexibility of\nPython-powered computational environments.", + "category": "nlin_CD" + }, + { + "text": "The smallest chimera states: We demonstrate that chimera behavior can be observed in small networks\nconsisting of three identical oscillators, with mutual all-to-all coupling.\nThree different types of chimeras, characterized by the coexistence of two\ncoherent oscillators and one incoherent oscillator (i.e. rotating with another\nfrequency) have been identified, where the oscillators show periodic (two\ntypes) and chaotic (one type) behaviors. Typical bifurcations at the\ntransitions from full synchronization to chimera states and between different\ntypes of chimeras have been described. Parameter regions for the chimera states\nare obtained in the form of Arnold tongues, issued from a singular parameter\npoint. Our analysis suggests that chimera states can be observed in small\nnetworks, relevant to various real-world systems.", + "category": "nlin_CD" + }, + { + "text": "On ergodicity for multi-dimensional harmonic oscillator systems with\n Nose-Hoover type thermostat: A simple proof and detailed analysis on the non-ergodicity for\nmultidimensional harmonic oscillator systems with Nose-Hoover type thermostat\nare given. The origin of the nonergodicity is symmetries in the\nmultidimensional target physical system, and is differ from that in the\nNose-Hoover thermostat with the 1-dimensional harmonic oscillator. A new simple\ndeterministic method to recover the ergodicity is also presented. An individual\nthermostat variable is attached to each degree of freedom, and all these\nvariables act on a friction coefficient for each degree of freedom. This action\nis linear and controlled by a Nos\\'e mass matrix Q, which is a matrix analogue\nof the scalar Nos\\'e's mass. Matrix Q can break the symmetry and contribute to\nattain the ergodicity.", + "category": "nlin_CD" + }, + { + "text": "Goos-Haenchen shift and localization of optical modes in deformed\n microcavities: Recently, an interesting phenomenon of spatial localization of optical modes\nalong periodic ray trajectories near avoided resonance crossings has been\nobserved [J. Wiersig, Phys. Rev. Lett. 97, 253901 (2006)]. For the case of a\nmicrodisk cavity with elliptical cross section we use the Husimi function to\nanalyse this localization in phase space. Moreover, we present a semiclassical\nexplanation of this phenomenon in terms of the Goos-Haenchen shift which works\nvery well even deep in the wave regime. This semiclassical correction to the\nray dynamics modifies the phase space structure such that modes can localize\neither on stable islands or along unstable periodic ray trajectories.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of traffic jams: order and chaos: By means of a novel variational approach we study ergodic properties of a\nmodel of a multi lane traffic flow, considered as a (deterministic) wandering\nof interacting particles on an infinite lattice. For a class of initial\nconfigurations of particles (roughly speaking satisfying the Law of Large\nNumbers) the complete description of their limit (in time) behavior is\nobtained, as well as estimates of the transient period. In this period the main\nobject of interest is the dynamics of `traffic jams', which is rigorously\ndefined and studied. It is shown that the dynamical system under consideration\nis chaotic in a sense that its topological entropy (calculated explicitly) is\npositive. Statistical quantities describing limit configurations are obtained\nas well.", + "category": "nlin_CD" + }, + { + "text": "Finite-size effects on open chaotic advection: We study the effects of finite-sizeness on small, neutrally buoyant,\nspherical particles advected by open chaotic flows. We show that, when\nprojected onto configuration space, the advected finite-size particles disperse\nabout the unstable manifold of the chaotic saddle that governs the passive\nadvection. Using a discrete-time system for the dynamics, we obtain an\nexpression predicting the dispersion of the finite-size particles in terms of\ntheir Stokes parameter at the onset of the finite-sizeness induced dispersion.\nWe test our theory in a system derived from a flow and find remarkable\nagreement between our expression and the numerically measured dispersion.", + "category": "nlin_CD" + }, + { + "text": "Dynamics of the Shapovalov mid-size firm model: One of the main tasks in the study of financial and economic processes is\nforecasting and analysis of the dynamics of these processes. Within this task\nlie important research questions including how to determine the qualitative\nproperties of the dynamics and how best to estimate quantitative indicators.\n These questions can be studied both empirically and theoretically. In the\nempirical approach, one considers the real data represented by time series,\nidentifies patterns of their dynamics, and then forecasts short- and long-term\nbehavior of the process. The second approach is based on postulating the laws\nof dynamics for the process, deriving mathematical dynamic models based on\nthese laws, and conducting subsequent analytical investigation of the dynamics\ngenerated by the models.\n To implement these approaches, both numerical and analytical methods can be\nused. It should be noted that while numerical methods make it possible to study\ncomplex models, the possibility of obtaining reliable results using them is\nsignificantly limited due to calculations being performed only over finite-time\nintervals, numerical errors, and the unbounded space of initial data sets. In\nturn, analytical methods allow researchers to overcome these problems and to\nobtain exact qualitative and quantitative characteristics of the process\ndynamics. However, their effective applications are often limited to\nlow-dimensional models. In this paper, we develop analytical methods for the\nstudy of deterministic dynamic systems. These methods make it possible not only\nto obtain analytical stability criteria and to estimate limiting behavior, but\nalso to overcome the difficulties related to implementing reliable numerical\nanalysis of quantitative indicators. We demonstrate the effectiveness of the\nproposed methods using the mid-size firm model suggested recently by V.I.\nShapovalov.", + "category": "nlin_CD" + }, + { + "text": "Dynamical epidemic suppression using stochastic prediction and control: We consider the effects of noise on a model of epidemic outbreaks, where the\noutbreaks appear. randomly. Using a constructive transition approach that\npredicts large outbreaks, prior to their occurrence, we derive an adaptive\ncontrol. scheme that prevents large outbreaks from occurring. The theory\ninapplicable to a wide range of stochastic processes with underlying\ndeterministic structure.", + "category": "nlin_CD" + }, + { + "text": "Finite-time synchronization of non-autonomous chaotic systems with\n unknown parameters: Adaptive control technique is adopted to synchronize two identical\nnon-autonomous systems with unknown parameters in finite time. A virtual\nunknown parameter is introduced in order to avoid the unknown parameters from\nappearing in the controllers and parameters update laws. The Duffing equation\nand a gyrostat system are chosen as the numerical examples to show the validity\nof the present method.", + "category": "nlin_CD" + }, + { + "text": "Amplitude distribution of eigenfunctions in mixed systems: We study the amplitude distribution of irregular eigenfunctions in systems\nwith mixed classical phase space. For an appropriately restricted random wave\nmodel a theoretical prediction for the amplitude distribution is derived and\ngood agreement with numerical computations for the family of limacon billiards\nis found. The natural extension of our result to more general systems, e.g.\nwith a potential, is also discussed.", + "category": "nlin_CD" + }, + { + "text": "Combined effects of compressibility and helicity on the scaling regimes\n of a passive scalar advected by turbulent velocity field with finite\n correlation time: The influence of compressibility and helicity on the stability of the scaling\nregimes of a passive scalar advected by a Gaussian velocity field with finite\ncorrelation time is investigated by the field theoretic renormalization group\nwithin two-loop approximation. The influence of helicity and compressibility on\nthe scaling regimes is discussed as a function of the exponents $\\epsilon$ and\n$\\eta$, where $\\epsilon$ characterizes the energy spectrum of the velocity\nfield in the inertial range $E\\propto k^{1-2\\epsilon}$, and $\\eta$ is related\nto the correlation time at the wave number $k$ which is scaled as\n$k^{-2+\\eta}$. The restrictions given by nonzero compressibility and helicity\non the regions with stable infrared fixed points which correspond to the stable\ninfrared scaling regimes are discussed. A special attention is paid to the case\nof so-called frozen velocity field when the velocity correlator is time\nindependent. In this case, explicit inequalities which must be fulfilled in the\nplane $\\epsilon-\\eta$ are determined within two-loop approximation.", + "category": "nlin_CD" + }, + { + "text": "Transition from homogeneous to inhomogeneous limit cycles: Effect of\n local filtering in coupled oscillators: We report an interesting symmetry-breaking transition in coupled identical\noscillators, namely the continuous transition from homogeneous to inhomogeneous\nlimit cycle oscillations. The observed transition is the oscillatory analog of\nthe Turing-type symmetry-breaking transition from amplitude death (i.e., stable\nhomogeneous steady state) to oscillation death (i.e., stable inhomogeneous\nsteady state). This novel transition occurs in the parametric zone of\noccurrence of rhythmogenesis and oscillation death as a consequence of the\npresence of local filtering in the coupling path. We consider paradigmatic\noscillators, such as Stuart-Landau and van der Pol oscillators under mean-field\ncoupling with low-pass or all-pass filtered self-feedback and through a\nrigorous bifurcation analysis we explore the genesis of this transition.\nFurther, we experimentally demonstrate the observed transition, which\nestablishes its robustness in the presence of parameter fluctuations and noise.", + "category": "nlin_CD" + }, + { + "text": "Community structure in real-world networks from a non-parametrical\n synchronization-based dynamical approach: This work analyzes the problem of community structure in real-world networks\nbased on the synchronization of nonidentical coupled chaotic R\\\"{o}ssler\noscillators each one characterized by a defined natural frequency, and coupled\naccording to a predefined network topology. The interaction scheme contemplates\nan uniformly increasing coupling force to simulate a society in which the\nassociation between the agents grows in time. To enhance the stability of the\ncorrelated states that could emerge from the synchronization process, we\npropose a parameterless mechanism that adapts the characteristic frequencies of\ncoupled oscillators according to a dynamic connectivity matrix deduced from\ncorrelated data. We show that the characteristic frequency vector that results\nfrom the adaptation mechanism reveals the underlying community structure\npresent in the network.", + "category": "nlin_CD" + }, + { + "text": "Quantitative predictions with detuned normal forms: The phase-space structure of two families of galactic potentials is\napproximated with a resonant detuned normal form. The normal form series is\nobtained by a Lie transform of the series expansion around the minimum of the\noriginal Hamiltonian. Attention is focused on the quantitative predictive\nability of the normal form. We find analytical expressions for bifurcations of\nperiodic orbits and compare them with other analytical approaches and with\nnumerical results. The predictions are quite reliable even outside the\nconvergence radius of the perturbation and we analyze this result using\nresummation techniques of asymptotic series.", + "category": "nlin_CD" + }, + { + "text": "Multistability in Piecewise Linear Systems by Means of the Eigenspectra\n Variation and the Round Function: A multistable system generated by a Piecewise Linear (PWL) system based on\nthe jerky equation is presented. The systems behaviour is characterised by\nmeans of the Nearest Integer or round(x) function to control the switching\nevents and to locate the corresponding equilibria among each of the commutation\nsurfaces. These surfaces are generated by means of the switching function\ndividing the space in regions equally distributed along one axis. The\ntrajectory of this type of system is governed by the eigenspectra of the\ncoefficient matrix which can be adjusted by means of a bifurcation parameter.\nThe behaviour of the system can change from multi-scroll attractors into a\nmono-stable state to the coexistence of several single-scroll attractors into a\nmulti-stable state. Numerical results of the dynamics and bifurcation analyses\nof their parameters are displayed to depict the multi-stable states.", + "category": "nlin_CD" + }, + { + "text": "Lyapunov exponent for inertial particles in the 2D Kraichnan model as a\n problem of Anderson localization with complex valued potential: We exploit the analogy between dynamics of inertial particle pair separation\nin a random-in-time flow and the Anderson model of a quantum particle on the\nline in a spatially random real-valued potential. Thereby we get an exact\nformula for the Lyapunov exponent of pair separation in a special case, and we\nare able to generalize the class of solvable models slightly, for potentials\nthat are real up to a global complex multiplier. A further important result for\ninertial particle behavior, supported by analytical computations in some cases\nand by numerics more generally, is that of the decay of the Lyapunov exponent\nwith large Stokes number (quotient of particle relaxation and flow turn-over\ntime-scales) as Stokes number to the power -2/3.", + "category": "nlin_CD" + }, + { + "text": "Langevin approach to synchronization of hyperchaotic time-delay dynamics: In this paper, we characterize the synchronization phenomenon of hyperchaotic\nscalar non-linear delay dynamics in a fully-developed chaos regime. Our results\nrely on the observation that, in that regime, the stationary statistical\nproperties of a class of hyperchaotic attractors can be reproduced with a\nlinear Langevin equation, defined by replacing the non-linear delay force by a\ndelta-correlated noise. Therefore, the synchronization phenomenon can be\nanalytically characterized by a set of coupled Langevin equations. We apply\nthis formalism to study anticipated synchronization dynamics subject to\nexternal noise fluctuations as well as for characterizing the effects of\nparameter mismatch in a hyperchaotic communication scheme. The same procedure\nis applied to second order differential delay equations associated to\nsynchronization in electro-optical devices. In all cases, the departure with\nrespect to perfect synchronization is measured through a similarity function.\nNumerical simulations in discrete maps associated to the hyperchaotic dynamics\nsupport the formalism.", + "category": "nlin_CD" + }, + { + "text": "Chaos in Nonlinear Random Walks with Non-Monotonic Transition\n Probabilities: Random walks serve as important tools for studying complex network\nstructures, yet their dynamics in cases where transition probabilities are not\nstatic remain under explored and poorly understood. Here we study nonlinear\nrandom walks that occur when transition probabilities depend on the state of\nthe system. We show that when these transition probabilities are non-monotonic,\ni.e., are not uniformly biased towards the most densely or sparsely populated\nnodes, but rather direct random walkers with more nuance, chaotic dynamics\nemerge. Using multiple transition probability functions and a range of networks\nwith different connectivity properties, we demonstrate that this phenomenon is\ngeneric. Thus, when such non-monotonic properties are key ingredients in\nnonlinear transport applications complicated and unpredictable behaviors may\nresult.", + "category": "nlin_CD" + }, + { + "text": "Bifurcation analysis and chaos control of periodically driven discrete\n fractional order memristive Duffing Oscillator: Discrete fractional order chaotic systems extends the memory capability to\ncapture the discrete nature of physical systems. In this research, the\nmemristive discrete fractional order chaotic system is introduced. The dynamics\nof the system was studied using bifurcation diagrams and phase space\nconstruction. The system was found chaotic with fractional order\n$0.465