discrete breathers in thermal equilibrium distributions and energy gaps we study a discrete two dimensional nonlinear system that allows for discrete breather solutions we perform a spectral analysis of the lattice dynamics at thermal equilibrium and use a cooling technique to measure the amount of breathers at thermal equilibrium our results confirm the existence of an energy threshold for discrete breathers the cooling method provides a novel way of measuring and analyzing discrete breather distribution properties in thermal equilibrium

linear approximation of transfer function with a pole of fractional power two methods for the linear approximation of a transfer function with a pole of fractional power are presented analog circuit models are developed and their frequency response curves and step response curves are compared it was found that the pade method gives a better approximation than wang and hsia s method within the frequency limit as specified

on the dynamics of the holstein model from the anticontinuous limit we consider the holstein model describing an electron interacting with a lattice of identical oscillators we remark that the on site system i e the system in which the interaction between the different sites of the lattice vanishes is integrable and anisocronous this allows us to apply some recent nekhoroshev type results to show that corresponding to the majority of initial data in which the electron probability is concentrated on a finite number of sites the electron probability distribution is approximatively constant for times growing exponentially with the inverse of the coupling parameter moreover for the same times the total energy of the oscillator system is approximatively constant

discrete breathers in a two dimensional fermi pasta ulam lattice using asymptotic methods we investigate whether discrete breathers are supported by a two dimensional fermi pasta ulam lattice a scalar one component two dimensional fermi pasta ulam lattice is shown to model the charge stored within an electrical transmission lattice a third order multiple scale analysis in the semi discrete limit fails since at this order the lattice equations reduce to the 2 1 dimensional cubic nonlinear schrodinger nls equation which does not support stable soliton solutions for the breather envelope we therefore extend the analysis to higher order and find a generalised 2 1 dimensional nls equation which incorporates higher order dispersive and nonlinear terms as perturbations we find an ellipticity criterion for the wave numbers of the carrier wave numerical simulations suggest that both stationary and moving breathers are supported by the system calculations of the energy show the expected threshold behaviour whereby the energy of breathers does em not go to zero with the amplitude we find that the energy threshold is maximised by stationary breathers and becomes arbitrarily small as the boundary of the domain of ellipticity is approached

graph characterization via ihara coefficients the novel contributions of this paper are twofold first we demonstrate how to characterize unweighted graphs in a permutation invariant manner using the polynomial coefficients from the ihara zeta function i e the ihara coefficients second we generalize the definition of the ihara coefficients to edge weighted graphs for an unweighted graph the ihara zeta function is the reciprocal of a quasi characteristic polynomial of the adjacency matrix of the associated oriented line graph since the ihara zeta function has poles that give rise to infinities the most convenient numerically stable representation is to work with the coefficients of the quasi characteristic polynomial moreover the polynomial coefficients are invariant to vertex order permutations and also convey information concerning the cycle structure of the graph to generalize the representation to edge weighted graphs we make use of the reduced bartholdi zeta function we prove that the computation of the ihara coefficients for unweighted graphs is a special case of our proposed method for unit edge weights we also present a spectral analysis of the ihara coefficients and indicate their advantages over other graph spectral methods we apply the proposed graph characterization method to capturing graph class structure and clustering graphs experimental results reveal that the ihara coefficients are more effective than methods based on laplacian spectra

localization and coherence in nonintegrable systems we study the irreversible dynamics of nonlinear nonintegrable hamiltonian oscillator chains approaching their statistical asymptotic states in systems constrained by more than one conserved quantity the partitioning of the conserved quantities leads naturally to localized and coherent structures if the phase space is compact the final equilibrium state is governed by entropy maximization and the coherent structures are stable lumps in systems where the phase space is not compact the coherent structures can be collapsed represented in phase space by a heteroclinic connection of some unstable saddle to infinity

tangent bifurcation of band edge plane waves dynamical symmetry breaking and vibrational localization abstract we study tangent bifurcation of band edge plane waves in nonlinear hamiltonian lattices the lattice is translationally invariant we argue for the breaking of permutational symmetry by the new bifurcated periodic orbits the case of two coupled oscillators is considered as an example for the perturbation analysis where the symmetry breaking can be traced using poincare maps next we consider a lattice and derive the dependence of the bifurcation energy on the parameters of the hamiltonian function in the limit of large system sizes a necessary condition for the occurence of the bifurcation is the repelling of the band edge plane wave from the linear spectrum with increasing energy we conclude that the bifurcated orbits will consequently exponentially localize in the configurational space

the notion of persistence applied to breathers in thermal equilibrium abstract we study the thermal equilibrium of nonlinear klein gordon chains at the limit of small coupling anticontinuum limit we show that the persistence distribution associated to the local energy density is a useful tool to study the statistical distribution of the so called thermal breathers mainly when the equilibrium is characterized by long lived pinned excitations in that case the distribution of persistence intervals turns out to be a power law we demonstrate also that this generic behaviour has a counterpart in the power spectra where the high frequencies domains nicely collapse if properly rescaled these results are also compared to nonlinear klein gordon chains with a soft nonlinearity for which the thermal breathers are rather mobile entities finally we discuss the possibility of a breather induced anomalous diffusion law and show that despite a strong slowing down of the energy diffusion there are numerical evidences for a normal asymptotic diffusion mechanism but with exceptionally small diffusion coefficients

discrete breathers in honeycomb fermi pasta ulam lattices we consider the two dimensional fermi pasta ulam lattice with hexagonal honeycomb symmetry which is a hamiltonian system describing the evolution of a scalar valued quantity subject to nearest neighbour interactions using multiple scale analysis we reduce the governing lattice equations to a nonlinear schrodinger nls equation coupled to a second equation for an accompanying slow mode two cases in which the latter equation can be solved and so the system decoupled are considered in more detail firstly in the case of a symmetric potential we derive the form of moving breathers we find an ellipticity criterion for the wavenumbers of the carrier wave together with asymptotic estimates for the breather energy the minimum energy threshold depends on the wavenumber of the breather we find that this threshold is locally maximised by stationary breathers secondly for an asymmetric potential we find stationary breathers which even with a quadratic nonlinearity generate no second harmonic component in the breather plots of all our findings show clear hexagonal symmetry as we would expect from our lattice structure finally we compare the properties of stationary breathers in the square triangular and honeycomb lattices

bifurcation analysis of the localized modes dynamics in lattices with saturable nonlinearity abstract the dynamics of localized modes in discrete media with saturable nonlinearity are investigated the stability of stationary bright solitons is discussed from the view point of the energy minimum principle and mapping analysis due to the cascade saturation mechanism a bifurcation from trapped to transversely moving localized mode is found for particular values of high power the bifurcation coincides with the existence of the almost perfect separatrix in the corresponding area preserving map in addition the definition of the peierls nabarro effective potential is reconsidered

two routes to chaos in the fractional lorenz system with dimension continuously varying abstract the object of this paper is to reveal the relation between dynamics of the fractional system and its dimension defined as a sum of the orders of all involved derivatives we take the fractional lorenz system as example and regard one or three of its orders as bifurcation parameters in this framework we compute the corresponding bifurcation diagrams via an optimal poincare section technique developed by us and find there exist two routes to chaos when its dimension increases from some values to 3 one is the process of cascaded period doubling bifurcations and the other is a crisis boundary crisis which occurs in the evolution of chaotic transient behavior we would like to point out that our investigation is the first to find out that a fractional differential equations fdes system can evolve into chaos by the crisis furthermore we observe rich dynamical phenomena in these processes such as two stage cascaded period doubling bifurcations chaotic transients and the transition from coexistence of three attractors to mono existence of a chaotic attractor these are new and interesting findings for fdes systems which to our knowledge have not been described before

simple statistical explanation for the localization of energy in nonlinear lattices with two conserved quantities the localization of energy in the discrete nonlinear schrodinger equation is explained with statistical methods the partition function and the entropy of the system are computed for low amplitude initial conditions detailed predictions for the long time solution are derived localized high amplitude excitations absorb a surplus of energy when they emerge as a by product of the production of entropy in the small fluctuations the thermodynamic interpretation of this process applies to many dynamical systems with two conserved quantities

synchronization of chaotic systems involving fractional operators of liouville caputo type with variable order abstract in this paper we propose a state observer based approach to synchronize variable order fractional vof chaotic systems in particular this work is focused on complete synchronization with a so called unidirectional master slave topology the master is described by a dynamical system in state space representation whereas the slave is described by a state observer the slave is composed of a master copy and a correction term which in turn is constituted of an estimation error and an appropriate gain that assures the synchronization the differential equations of the vof chaotic system are described by the liouville caputo and atangana baleanu caputo derivatives numerical simulations involving the synchronization of rossler oscillators chua s systems and multi scrolls are studied the simulations show that different chaotic behaviors can be obtained if different smooths functions defined in the interval 0 1 are used as the variable order of the fractional derivatives furthermore simulations show that the vof chaotic systems can be synchronized

linear stability of perturbed hamiltonian systems theory and a case example in this paper we present a general theory based on a lyapunov schmidt reduction for the linearized stability of a perturbed hamiltonian system with a number of symmetries the reduction leads to an eigenvalue problem in a small subspace of the original system whose eigenvalues and eigenvectors can be used to derive the ones of the perturbed original system the krein signature of the eigenvalues can also be obtained in this framework the method is applied to the case example of the coupled nonlinear schr dinger equations giving excellent agreement with the full numerical linear stability results

design of a chatter free terminal sliding mode controller for nonlinear fractional order dynamical systems this paper investigates the problem of robust control of nonlinear fractional order dynamical systems in the presence of uncertainties first a novel switching surface is proposed and its finite time stability to the origin is proved subsequently using the sliding mode theory a robust fractional control law is proposed to ensure the existence of the sliding motion in finite time we use a fractional lyapunov stability theory to prove the stability of the system in a given finite time in order to avoid the chattering which is inherent in conventional sliding mode controllers we transfer the sign function of the control input into the fractional derivative of the control signal the proposed chattering free sliding mode technique is then applied for stabilisation of a broad class of three dimensional fractional order chaotic systems via a single variable driving control input simulation results reveal that the proposed fractional sliding mode controller works well for chaos control of fractional ord

depth based complexity traces of graphs in this paper we aim to characterize graphs in terms of a structural measure of complexity our idea is to decompose a graph into layered substructures of increasing size and then to measure the information content of these substructures to locate dominant substructures within a graph we commence by identifying a centroid vertex which has the minimum shortest path length variance to the remaining vertices for each graph a family of centroid expansion subgraphs is derived from the centroid vertex in order to capture dominant structural characteristics of the graph since the centroid vertex is identified through a global analysis of the shortest path length distribution the expansion subgraphs provide a fine representation of a graph structure we then show how to characterize graphs using depth based complexity traces here we explore two different strategies the first strategy is to measure how the entropies on the centroid expansion subgraphs vary with the increasing size of the subgraphs the second strategy is to measure how the entropy differences vary with the increasing size of the subgraphs we perform graph classification in the principal component space of the complexity trace vectors experiments on graph datasets abstracted from some bioinformatics and computer vision databases demonstrate the effectiveness and efficiency of the proposed graph complexity traces our methods are competitive to state of the art methods highlightswe identify a centroid vertex of a graph through its shortest path matrix we establish a family of expansion subgraphs of a graph from its centroid vertex we construct depth based complexity traces of a graph on its expansion subgraphs we evaluate the performance of our complexity traces on standard graph datasets we demonstrate the effectiveness and efficiency of the graph complexity traces

thermodynamic depth of causal states objective complexity via minimal representations thermodynamic depth is an appealing but flawed structural complexity measure it depends on a set of macroscopic states for a system but neither its original introduction by lloyd and pagels nor any follow up work has considered how to select these states depth therefore is at root arbitrary computational mechanics an alternative approach to structural complexity provides a definition for a system close quote s minimal necessary causal states and a procedure for finding them we show that the rate of increase in thermodynamic depth or ital dive is the system close quote s reverse time shannon entropy rate and so depth only measures degrees of macroscopic randomness not structure to fix this we redefine the depth in terms of the causal state representation emdash epsilon machines emdash and show that this representation gives the minimum dive consistent with accurate prediction thus epsilon machines are optimally shallow copyright ital 1999 ital the american physical society

localized breather like solution in a discrete klein gordon model and application to dna abstract non propagating localized oscillating solutions are studied in the case of a discrete one dimensional system with the 2 3 power onsite potential a multiple scale asymptotic perturbation expansion is not adequate for describing the large amplitude oscillations we consider starting from an exact pulse like solution in the continuum we derive an approximate oscillating solution our numerical simulations reveal that whereas the excitations are strongly damped in the strong intersite coupling case they are long lived in the discrete cases an additional modulation phenomenon of the oscillation seen in the simulation is explained by combination of the breather oscillations with the zero group velocity phonons modes then applying the green s functions formalism in analogy with a previous work s takeno et al prog theor phys suppl 94 1988 we derive a very accurate expression for the discrete breather with a quasi infinite lifetime lastly we demonstrate how such excitations can explain why the breathers make the dominant contributions in the dynamics of a simple model of dna

chaos and hyperchaos in the fractional order rossler equations abstract the dynamics of fractional order systems have attracted increasing attentions in recent years in this paper we numerically study the chaotic behaviors in the fractional order rossler equations we found that chaotic behaviors exist in the fractional order rossler equation with orders less than 3 and hyperchaos exists in the fractional order rossler hyperchaotic equation with order less than 4 the lowest orders we found for chaos and hyperchaos to exist in such systems are 2 4 and 3 8 respectively period doubling routes to chaos in the fractional order rossler equation are also found

q breathers in fermi pasta ulam chains existence localization and stability the fermi pasta ulam fpu problem consists of the nonequipartition of energy among normal modes of a weakly anharmonic atomic chain model in the harmonic limit each normal mode corresponds to a periodic orbit in phase space and is characterized by its wave number q we continue normal modes from the harmonic limit into the fpu parameter regime and obtain persistence of these periodic orbits termed here q breathers qb they are characterized by time periodicity exponential localization in the q space of normal modes and linear stability up to a size dependent threshold amplitude trajectories computed in the original fpu setting are perturbations around these exact qb solutions the qb concept is applicable to other nonlinear lattices as well

existence and stability of breathers in lattices of weakly coupled two dimensional near integrable hamiltonian oscillators a method of proving the existence of breathers in lattices of two dimensional weakly coupled hamiltonian oscillators is presented this method has the advantage that provides a first approximation of the initial conditions of the breather solution together with its linear stability since the proof relies on the use of action angle variables we also present a method to perform the calculations without knowledge of the specific transformation we illustrate the results in a lattice of quartic oscillators and in another one that consists of coupled 2d morse oscillators finally we examine the possibility of extending these results in lattices of more than two degrees of freedom per lattice site

nonlinear chains inside walls abstract the conservative dynamics of a 1d chain of units coupled with fpu type nonlinear interactions is considered stationary patterns in such chains emerge due to a balance of coupling energy between neighbouring units particularly interesting are the nontrivial stationary states which contain segments of positive and negative slope this results in a zig zag pattern in the case of periodic boundary conditions and in kink or anti kink solutions in the case of the free boundary conditions imposing constraints on the chain by way of two confining infinitely high walls has repercussions for the stability of these stationary states here such stationary states commensurable with the available space between the two walls are examined in detail and their respective stability properties are determined analytically by invoking the transfer matrix method strikingly stationary anti kink solutions and periodic zig zag states being unstable in the absence of confining walls become stable when confining walls are introduced furthermore simulations reveal that chains with randomly generated initial conditions can seek out these patterns thus localising energy and persist for considerable time

nonpersistence of breather families for the perturbed sine gordon equation we show that up to one exception and as a consequence of first order perturbation theory only it is impossible that a large portion of the well known family of breather solutions to the sine gordon equation could persist under any nontrivial perturbation of the form u tt u xx sin u varepsilon delta left u right o left varepsilon 2 right where is an analytic function in anarbitrarily small neighbourhood ofu 0 improving known results we analyze and overcome the particular difficulties that arise when one allows the domain of analyticity of to be small the single exception is a one dimensional linear space of perturbation functions under which the full family of breathers does persist up to first order in e

weisfeiler lehman graph kernels in this article we propose a family of efficient kernels for large graphs with discrete node labels key to our method is a rapid feature extraction scheme based on the weisfeiler lehman test of isomorphism on graphs it maps the original graph to a sequence of graphs whose node attributes capture topological and label information a family of kernels can be defined based on this weisfeiler lehman sequence of graphs including a highly efficient kernel comparing subtree like patterns its runtime scales only linearly in the number of edges of the graphs and the length of the weisfeiler lehman graph sequence in our experimental evaluation our kernels outperform state of the art graph kernels on several graph classification benchmark data sets in terms of accuracy and runtime our kernels open the door to large scale applications of graph kernels in various disciplines such as computational biology and social network analysis

discrete breathers in a two dimensional spring mass lattice we consider a two dimensional spring mass lattice with square symmetry in which each particle experiences a nonlinear onsite potential and nonlinear nearest neighbour interactions at equilibrium the particles are equally spaced in both the horizontal and vertical directions and all springs are unextended motivated by the work of marin et al 1998 phys lett a 248 225 9 2001 phys lett a 281 21 5 we seek a solution in which most of the breather s energy is focused along three chains we construct an asymptotic approximation to the breather using the method of multiple scales to describe the coherent oscillations in the three main chains that constitute the discrete breather we reduce the equation of motion to a nonlinear schrodinger equation for the leading order term and find a family of solutions which encompasses both stationary and moving bright soliton solutions we use numerical simulations of the lattice to verify the shape and velocity of breathers and find that while stationary breathers are found to persist for long times moving breathers decay by radiating energy in the direction perpendicular to their motion

existence and stability of quasi periodic breathers in networks of ginzburg landau oscillators abstract it is shown that there is an attracting quasi periodic breather in networks of ginzburg landau oscillators

fractional market dynamics a new extension of a fractality concept in nancial mathematics has been developed we have introduced a new fractional langevin type stochastic dierential equation that diers from the standard langevin equation i by replacing the rst order derivative with respect to time by the fractional derivative of order and ii by replacing white noise gaussian stochastic force by the generalized shot noise each pulse of which has a random amplitude with the stable levy distribution as an application of the developed fractional non gaussian dynamical approach the expression for the probability distribution function pdf of the returns has been established it is shown that the obtained fractional pdf ts well the central part and the tails of the empirical distribution of s p 500 returns c 2000 elsevier science b v all rights reserved

breathers in nonlinear lattices existence linear stability and quantization abstract it has been proved by the principle of anti integrable limit renamed here anti continuous limit and the implicit function theorem that under rather weak hypotheses there exist spatially localized and time periodic solutions breathers in arrays of nonlinear coupled classical oscillators provided the coupling is not too strong the models can be translationally invariant or not and be of any dimension there are also time periodic solutions corresponding to arbitrary distributions on the lattice of such solutions multibreathers the whole set of these solutions can be labeled by a discrete coding sequence the condition that the frequency of the breather or multibreather solution has no harmonics in the phonon spectrum can be discarded for a large subset of spatially extended time periodic solutions called phonobreather which correspond to special codes which can be also spatially chaotic and where no oscillator is at rest these results extend almost without change to models with coupled rotors the existence of rotobreathers which are exact solutions consisting of a single rotor rotating while the other rotors oscillate is proved there are also multirotobreather solutions and phono rotobreathers etc it is also shown that the phase of the single breathers constituting a multibreather or multirotobreather solution can be submitted to a phase torsion these new dynamical structures carry a nonzero energy flow in models with two dimensions and more the phase torsion of multibreather solutions can generate new dynamical solutions with vortices in the energy flow the dynamics of a quantum electron coupled to a classical lattice is also studied within the same approach from the anti continuous limit the existence of two kinds of polarobreathers which are spatially exponentially localized dynamical solutions is proved for a small enough electronic transfer integral the polarobreathers of the first kind include the standard static polaron the lattice configuration is static time independent when the electron is in an excited state i e the electronic oscillator oscillates periodically they can be spatially chaotic a polarobreather of the second kind corresponds to a localized periodic lattice oscillation associated with a localized quasiperiodic electronic oscillation the most simple of these solutions are those for which both the lattice oscillation and the electron wave function are mostly concentrated close to the same single site the standard floquet analysis of the linear stability of the breather and multibreather solutions is shown to be related with the band spectrum property of the newton operator involved in the implicit function theorem the krein theory of bifurcations is reinterpreted from this point of view the linear stability of the single breather and rotobreather solutions is then proved at least at small enough coupling finally it is shown that in the quantum version of translationally invariant classical models where there cannot exist in principle any strictly localized excitation there are narrow bands of excitations consisting of bound states of many bosons which are the quantum counterpart of the classical breathers the bandwidth of these excitations goes exponentially to zero with the number of bound phonons this theory provides simultaneously existence theorems and a new powerful and accurate numerical method for calculating practically any of the solutions which has been formally predicted to exist these numerical applications are currently written elsewhere

fractal system as represented by singularity function a fractional slope on the log log bode plot has been observed in characterizing a certain type of physical phenomenon and has been called the fractal system or the fractional power pole in order to represent and study its dynamical behavior a singularity function method is presented which consists of cascaded branches of a number of pole zero negative real pairs or simple rc section the distribution spectrum of the system can also be easily calculated and its accuracy depends on a prescribed error specified in the beginning the method is then extended to a multiple fractal system which consists of a number of fractional power poles the result can be simulated by a combination of singularity functions each representing a single fractal system

backtrackless walks on a graph the aim of this paper is to explore the use of backtrackless walks and prime cycles for characterizing both labeled and unlabeled graphs the reason for using backtrackless walks and prime cycles is that they avoid tottering and can increase the discriminative power of the resulting graph representation however the use of such methods is limited in practice because of their computational cost in this paper we present efficient methods for computing graph kernels which are based on backtrackless walks in a labeled graph and whose worst case running time is the same as that of kernels based on random walks for clustering unlabeled graphs we construct feature vectors using ihara coefficients since these coefficients are related to the frequencies of prime cycles in the graph to efficiently compute the low order coefficients we present an o v 3 algorithm which is better than the o v 6 worst case running time of previously known algorithms in the experimental evaluation we apply the proposed method to clustering both labeled and unlabeled graphs the results show that using backtrackless walks and prime cycles instead of random walks can increase the accuracy of recognition

numerical study of two dimensional disordered klein gordon lattices with cubic soft anharmonicity localized oscillations appear both in ordered nonlinear lattices breathers and in disordered linear lattices anderson modes numerical studies on a class of two dimensional systems of the klein gordon type show that there exist two different types of bifurcation in the path from nonlinearity order to linearity disorder inverse pitchforks with or without period doubling and saddle nodes this was discovered for a one dimensional system in a previous work of archilla mackay and marin the appearance of a saddle node bifurcation indicates that nonlinearity and disorder begin to interfere destructively and localization is not possible in contrast the appearance of a pitchfork bifurcation indicates that localization persists

graph theoretical constructions for graph entropy and network coding based communications the guessing number of a directed graph digraph equivalent to the entropy of that digraph was introduced as a direct criterion on the solvability of a network coding instance this paper makes two contributions on the guessing number first we introduce an undirected graph on all possible configurations of the digraph referred to as the guessing graph which encapsulates the essence of dependence amongst configurations we prove that the guessing number of a digraph is equal to the logarithm of the independence number of its guessing graph therefore network coding solvability is no more a problem on the operations made by each node but is simplified into a problem on the messages that can transit through the network by studying the guessing graph of a given digraph and how to combine digraphs or alphabets we are thus able to derive bounds on the guessing number of digraphs second we construct specific digraphs with high guessing numbers yielding network coding instances where a large amount of information can transit we first propose a construction of digraphs with finite parameters based on cyclic codes with guessing number equal to the degree of the generator polynomial we then construct an infinite class of digraphs with arbitrary girth for which the ratio between the linear guessing number and the number of vertices tends to one despite these digraphs being arbitrarily sparse these constructions yield solvable network coding instances with a relatively small number of intermediate nodes for which the node operations are known and linear although these instances are sparse and the sources are arbitrarily far from their corresponding sinks

discrete breathers classical and quantum abstract discrete breathers are time periodic spatially localised motions in networks of oscillators their occurrence in three principal contexts is reviewed i autonomous hamiltonian or reversible systems ii autonomous forced damped systems and iii time periodically forced systems the main proposed area of application however is to molecular crystals where quantum effects must be taken into account thus an idea for a theory of quantum discrete breathers is also presented though some details remain to be worked out

discrete compactons some exact results in this letter we use the method of constructing exact solutions on lattices proposed by kinnersley and described in schmidt 1979 phys rev b 20 4397 to obtain exact compacton solutions in discrete models we examine the linear stability of such solutions both for the bright compacton and for the dark compacton cases we focus on a quantization condition that the width of the profile should satisfy we also use this quantization condition to examine the possibility of compact coherent structures travelling in discrete settings our results are obtained for sinusoidal profiles and then generalized to elliptic functions of arbitrary modulus the possibility of multi compacton solutions is considered

nonlinear localized periodic solutions in a coupled map lattice abstract we prove the existence of nonlinear localized time periodic solutions in a chain of symplectic mappings with nearest neighbour coupling this is a class of systems whose behaviour can be seen as a representation of a lattice of pendula the effect of discrete time changes the mathematical as well as the numerical procedures applying the discrete version of floquet theory eases and clarifies the procedure of proving the existence of the localized time periodic solutions as an extension of the concept of rotobreathers one can produce solutions which rotate at every site of the lattice to consider these we use a general definition of localization

nonlinear observer design to synchronize fractional order chaotic systems via a scalar transmitted signal abstract in this paper based on the idea of nonlinear observer and stability theory of fractional order systems a new systematic scheme to synchronize a class of fractional order chaotic systems via a scalar transmitted signal is developed the approach is simple global and theoretically rigorous it enables synchronization of fractional order chaotic systems to be achieved in a systematic way and does not require the computation of the conditional lyapunov exponents simulation results are used to visualize and illustrate the effectiveness of the proposed synchronization method

nonlinear modes in coupled rotator models abstract we compare the properties of generalized nonlinear klein gordon equations with a sinusoidal coupling sine lattice equations to those of the usual nonlinear klein gordon equation with harmonic coupling for three equations leading to the same sine gordon equation in the continuum limit using hirota s bilinear method we find a strong analogy between the properties of nonlinear oscillatory modes in the three equations numerical simulations confirm this analogy even for very discrete cases however we point out a fundamental difference between the harmonic and sinusoidal coupling by exhibiting a new class of localized nonlinear excitations which are specific to the sinusoidal coupling the rotating modes they can be created thermally an approximate analytical solution is obtained for these modes which are intrinsically discrete and appear to be extremely stable even in collisions with breathers possible implications for physics are discussed

chaotic trajectories in the standard map the concept of anti integrability a rigorous proof is given in the standard map associated with a frenkel kontorowa model for the existence of chaotic trajectories with unbounded momenta for large enough coupling constant k k0 these chaotic trajectories with finite entropy per site are coded by integer sequences mi such that the sequence bi mi 1 mi 1 2mi be bounded by some integer b the bound k0 in k depends on b and can be lowered for coding sequences mi fulfilling more restrictive conditions the obtained chaotic trajectories correspond to stationary configurations of the frenkel kontorowa model with a finite non zero photon gap called gap parameter in dimensionless units this property implies that the trajectory or the configuration ui can be uniquely continued as a uniformly continuous function of the model parameter k in some neighborhood of the initial configuration a non zero gap parameter implies that the lyapunov coefficient is strictly positive when it is defined in addition the existence of dilating and contracting manifolds is proven for these chaotic trajectories exotic trajectories such as ballistic trajectories are also proven to exist as a consequence of these theorems the concept of anti integrability emerges from these theorems in the anti integrable limit which can be only defined for a discrete time dynamical system the coordinates of the trajectory at time i do not depend on the coordinates at time i 1 thus at this singular limit the existence of chaotic trajectories is trivial and the dynamical system reduces to a bernoulli shift it is well known that the kam tori of symplectic dynamical originates by continuity from the invariant tori which exists in the integrible limit under certain conditions in a similar way it appears that the chaotic trajectories of dynamical systems originate by continuity from those which exists at the anti integrable limits also under certain conditions

on the variational structure of breather solutions i sine gordon equation abstract in this paper we describe stability properties of the sine gordon breather solution these properties are first described by suitable variational elliptic equations which also implies that the stability problem reduces in some sense to i the study of the spectrum of explicit linear systems and ii the understanding of how bad directions if any can be controlled using low regularity conservation laws then we discuss spectral properties of a fourth order linear matrix system using numerical methods we confirm that all spectral assumptions leading to the h 2 h 1 stability of sg breathers are numerically satisfied even in the ultra relativistic singular regime

graph matching using the interference of continuous time quantum walks we consider how continuous time quantum walks can be used for graph matching we focus in detail on both exact and inexact graph matching and consider in depth the problem of measuring graph similarity we commence by constructing an auxiliary graph in which the two graphs to be matched are co joined by a layer of indicator vertices one for each potential correspondence between a pair of vertices we simulate a continuous time quantum walk in parallel on the two graphs the layer of connecting indicator vertices in the auxiliary graph allow quantum interference to take place between the two walks the interference amplitudes on the indicator vertices are determined by differences in the two walks and can be used to calculate probabilities for matches between pairs of vertices from the graphs by applying the hungarian kuhn munkres algorithm to these probabilities we recover a correspondence mapping between the graphs to calculate graph similarity we combine these probabilities with edge consistency information to give a consistency measure based on the consistency measure we define two graph similarity measures one of which requires correspondence matches while the second does not we analyse our approach experimentally using synthetic and real world graphs this reveals that our method gives results that are intermediate between the most sophisticated iterative techniques available and simpler less complex ones

targeted energy transfer between a rotor and a morse oscillator a model for selective chemical dissociation abstract standard kramers theory of chemical reactions involves a coupling with a langevin thermal bath which intrinsically forbids the possible existence of discrete breathers dbs i e local modes however it is now known that in complex systems that energy may focus for long time as discrete breathers local mode in very special systems targeted energy transfer may occur subsequently to another selected site and induces an ultraselective chemical reaction operating at low temperature the dynamics of the reaction is non brownian but highly coherent along a specific path in the phase space where the system is nearly integrable chemical expressway a simple toy model illustrating this idea is reduced to a rotor weakly coupled to a morse oscillator supposed to represent two specific local modes in a complex system which are appropriately tuned for targeted energy transfer when the rotor is initially rotating with a frequency resonant with those of the morse oscillator at rest the energy of the rotor is almost completely transferred to the morse oscillator and induces chemical dissociation the periodic oscillations of the rotor and morse oscillator remain coherent and their frequencies simultaneously vary but always remain resonant this process is analytically described within an integrable approximation numerical investigations of this model confirm that in the appropriate conditions the particle in the morse oscillator is indeed promptly ejected at infinity with a finite velocity chemical dissociation despite some chaotic transient manifesting imperfect integrability

localized excitations in two dimensional hamiltonian lattices we analyze the origin and features of localized excitations in a discrete two dimensional hamiltonian lattice the lattice obeys discrete translational symmetry and the localized excitations exist because of the presence of nonlinearities we connect the presence of these excitations with the existence of local integrability of the original n degree of freedom system on the basis of this explanation we make several predictions about the existence and stability of these excitations this work is an extension of previously published results on vibrational localization in one dimensional nonlinear hamiltonian lattices phys rev e 49 1994 836 thus we confirm earlier suggestions about the generic property of hamiltonian lattices to exhibit localized excitations independent on the dimensionality of the lattice

bound states of two dimensional solitons in the discrete nonlinear schrodinger equation the existence and stability of bound states bss of solitary excitations in the two dimensional discrete nonlinear schrodinger lattice are considered stable bss of two or three solitons are categorized according to their vorticity s s 0 corresponds to ordinary pulses while s 1 corresponds to discrete vortex solitons interactions of s 0 solitons are found to have clear particle like characteristics which can be very accurately predicted in the framework of the perturbation theory that we develop triangular bound states of three solitons are also constructed and particle like characteristics in their behaviour are followed in numerical experiments finally stable bound states of two vortex solitons and s 0 1 stable complexes are constructed for the latter state we find a weak dependence of the interaction generated stability eigenvalue on the distance between the two solitons

fast depth based subgraph kernels for unattributed graphs in this paper we investigate two fast subgraph kernels based on a depth based representation of graph structure both methods gauge depth information through a family of k layer expansion subgraphs rooted at a vertex 1 the first method commences by computing a centroid based complexity trace for each graph using a depth based representation rooted at the centroid vertex that has minimum shortest path length variance to the remaining vertices 2 this subgraph kernel is computed by measuring the jensen shannon divergence between centroid based complexity entropy traces the second method on the other hand computes a depth based representation around each vertex in turn the corresponding subgraph kernel is computed using isomorphisms tests to compare the depth based representation rooted at each vertex in turn for graphs with n vertices the time complexities for the two new kernels are o n 2 and o n 3 in contrast to o n 6 for the classic gartner graph kernel 3 key to achieving this efficiency is that we compute the required shannon entropy of the random walk for our kernels with o n 2 operations this computational strategy enables our subgraph kernels to easily scale up to graphs of reasonably large sizes and thus overcome the size limits arising in state of the art graph kernels experiments on standard bioinformatics and computer vision graph datasets demonstrate the effectiveness and efficiency of our new subgraph kernels highlightswe investigate two fast subgraph kernels based on a depth based representation our kernels gauge depth information rooted at a vertex for a graph the time complexities for the two kernels are o n 2 and o n 3 we evaluate the performance of our subgraph kernels on standard graph datasets we demonstrate the effectiveness of the proposed subgraph kernels

on a class of discretizations of hamiltonian nonlinear partial differential equations abstract we present a new class of discretizations of partial differential equations pdes that preserve a momentum like integral of the motion this results in an effective translational invariance for the dynamical problem and the absence of a peierls nabarro barrier that is usually present in discretizations of hamiltonian pdes a general method to construct such discretizations for any nonlinearity is given and the properties of the resulting differential difference equations are analyzed in a number of different cases for nonlinear klein gordon as well for nonlinear schrodinger type systems while for the former nonintegrability of the dynamical problem is evident in the latter case numerical evidence suggests that the behavior is close to the one of integrable systems we also show how static solutions of the equations can be constructed for these discretizations and discuss the similarities and differences of the method with previously reported ones

discrete nonlinear schrodinger equations with superlinear nonlinearities abstract in this paper we study the discrete nonlinear equation lu n u n n g n u n n z where belongs to a finite spectral gap of the operator l or it is a lower edge of a finite spectral gap by using a variant generalized weak linking theorem for strongly indefinite problem developed by schechter and zou we will study the existence of nontrivial solitons for this equation with gn is super linear

global existence and compact attractors for the discrete nonlinear schrodinger equation abstract we study the asymptotic behavior of solutions of discrete nonlinear schrodinger type dnls equations for a conservative system we consider the global in time solvability and the question of existence of standing wave solutions similarities and differences with the continuous counterpart nls partial differential equation are pointed out for a dissipative system we prove existence of a global attractor and its stability under finite dimensional approximations similar questions are treated in a weighted phase space finally we propose possible extensions for various types of dnls equations

discrete breathers in vibroimpact chains analytic solutions we present exact analytic solutions for discrete breathers in essentially nonlinear oscillatory chains belonging to both of the most common universality classes klein gordon and fermi pasta ulam the exact solutions can be obtained due to use of vibroimpact potentials combining extreme nonlinearity with the possibility of description in terms of a forced linear model under conditions of self consistency a crossover between the cases of high and low energies can be studied directly the solutions obtained may be used as a high energy limit for models with other realistic potentials as well as benchmarks for the testing of approximate approaches in the theory of discrete breathers

conditions on the existence of localized excitations in nonlinear discrete systems we use recent results that localized excitations in nonlinear hamiltonian lattices can be viewed and described as multiple frequency excitations their dynamics in phase space takes place on tori of corresponding dimension for a one dimensional hamiltonian lattice with nearest neighbour interaction we transform the problem of solving the coupled differential equations of motion into a certain mapping m l 1 f m l m l 1 where m l for every l lattice site is a function defined on an infinite discrete space of the same dimension as the torus we consider this mapping in the tails of the localized excitation i e for l rightarrow pm infty for a generic hamiltonian lattice the thus linearized mapping is analyzed we find conditions of existence of periodic one frequency localized excitations as well as of multiple frequency excitations the symmetries of the solutions are obtained as a result we find that the existence of localized excitations can be a generic property of nonlinear hamiltonian lattices in contrast to nonlinear hamiltonian fields

stable and metastable states and the formation and destruction of breathers in the discrete nonlinear schrodinger equation statistical mechanics explains many localization phenomena of lattices such as the discrete nonlinear schrodinger equation however numerical simulations show that the complete thermalization is rarely achieved instead one observes metastable statistical states that are robust when excited locally this paper investigates thermodynamically metastable states where the trajectory is confined to some part of the energy shell the partition function and the entropy are computed with a perturbation method this method is applicable to stable and metastable states and it allows us to give approximative analytic expressions for the entropy in the complete thermodynamic state space

equivalence of uniform hyperbolicity for symplectic twist maps and phonon gap for frenkel kontorova models there is a well known correspondence between the dynamics of symplectic twist maps which represent an important class of hamiltonian systems and the equilibrium states of a class of variational problems in solid state physics known as frenkel kontorova models in this paper it is shown that the key concepts of uniform hyperbolicity in the first context and phonon gap in the second context are equivalent this allows one to transfer many ideas between the two and hence to deduce new results for example we prove the uniform hyperbolicity of certain invariant sets for symplectic twist maps constructed from so called non degenerate anti integrable limits by aubry and abramovici using the concept of phonon gap and deduce that they have measure zero we prove existence of many multi defect equilibrium states for a frenkel kontorova system if it has a single non degenerate defect equilibrium state via standard results in the theory of uniform hyperbolicity

large memory capacity in chaotic artificial neural networks a view of the anti integrable limit in the literature it was reported that the chaotic artificial neural network model with sinusoidal activation functions possesses a large memory capacity as well as a remarkable ability of retrieving the stored patterns better than the conventional chaotic model with only monotonic activation functions such as sigmoidal functions this paper from the viewpoint of the anti integrable limit elucidates the mechanism inducing the superiority of the model with periodic activation functions that includes sinusoidal functions particularly by virtue of the anti integrable limit technique this paper shows that any finite dimensional neural network model with periodic activation functions and properly selected parameters has much more abundant chaotic dynamics that truly determine the model s memory capacity and pattern retrieval ability to some extent this paper mathematically and numerically demonstrates that an appropriate choice of the activation functions and control scheme can lead to a large memory capacity and better pattern retrieval ability of the artificial neural network models

a quantum jensen shannon graph kernel for unattributed graphs in this paper we use the quantum jensen shannon divergence as a means of measuring the information theoretic dissimilarity of graphs and thus develop a novel graph kernel in quantum mechanics the quantum jensen shannon divergence can be used to measure the dissimilarity of quantum systems specified in terms of their density matrices we commence by computing the density matrix associated with a continuous time quantum walk over each graph being compared in particular we adopt the closed form solution of the density matrix introduced in rossi et al 2013 27 28 to reduce the computational complexity and to avoid the cumbersome task of simulating the quantum walk evolution explicitly next we compare the mixed states represented by the density matrices using the quantum jensen shannon divergence with the quantum states for a pair of graphs described by their density matrices to hand the quantum graph kernel between the pair of graphs is defined using the quantum jensen shannon divergence between the graph density matrices we evaluate the performance of our kernel on several standard graph datasets from both bioinformatics and computer vision the experimental results demonstrate the effectiveness of the proposed quantum graph kernel highlightswe compute a density matrix for a graph using the continuous time quantum walk we compute the quantum jensen shannon divergence between graph density matrixes we define a quantum jensen shannon graph kernel using the quantum divergence we evaluate the performance of our quantum kernel on standard graph datasets we demonstrate the effectiveness of the proposed quantum kernel

obtaining breathers in nonlinear hamiltonian lattices we present a numerical method for obtaining high accuracy numerical solutions of spatially localized time periodic excitations on a nonlinear hamiltonian lattice we compare these results with analytical considerations of the spatial decay we show that nonlinear contributions have to be considered and we obtain very good agreement between the latter and numerical results we discuss further applications of the method and results

shannon and von neumann entropy of random networks with heterogeneous expected degree entropic measures of complexity are able to quantify the information encoded in complex network structures several entropic measures have been proposed in this respect here we study the relation between the shannon entropy and the von neumann entropy of networks with given expected degree sequence we find in different examples of network topologies that when the degree distribution contains some heterogeneity an intriguing correlation emerges between the two entropic quantities this results seems to suggest that heterogeneity in the expected degree distribution is implying an equivalence between a quantum and a classical description of networks which respectively corresponds to the von neumann and the shannon entropy

discrete breathers in a two dimensional hexagonal fermi pasta ulam lattice we consider a two dimensional fermi pasta ulam fpu lattice with hexagonal symmetry using asymptotic methods based on small amplitude ansatz at third order we obtain a eduction to a cubic nonlinear schr o dinger equation nls for the breather envelope however this does not support stable soliton solutions so we pursue a higher order analysis yielding a generalised nls which includes known stabilising terms we present numerical results which suggest that long lived stationary and moving breathers are supported by the lattice we find breather solutions which move in an arbitrary direction an ellipticity criterion for the wavenumbers of the carrier wave symptotic estimates for the breather energy and a minimum threshold energy below which breathers cannot be found this energy threshold is maximised for stationary breathers and becomes vanishingly small near the boundary of the elliptic domain where breathers attain a maximum speed several of the results obtained are similar to those obtained for the square fpu lattice butt wattis em j phys a bf 39 4955 2006 though we find that the square and hexagonal lattices exhibit different properties in regard to the generation of harmonics and the isotropy of the generalised nls equation

chaotic behavior in noninteger order cellular neural networks in this paper a simple system showing chaotic behavior is introduced it is based on the well known concept of cellular neural networks cnns which have already given good results in generating complex dynamics the peculiarity of the cnn model consists in the fact that it replaces the traditional first order cell with a noninteger order one the introduction of the fractional cell with a suitable choice of the coupling parameters leads to the onset of chaos in a simple two cell system a theoretical approach based on the harmonic balance theory has been used to investigate the existence of chaos a circuit realization of the proposed fractional two cell chaotic cnn is reported and the corresponding strange attractor is also shown

standing waves for discrete schrodinger equations in infinite lattices with saturable nonlinearities abstract in this paper we consider the periodic discrete nonlinear equation l u n u n g n u n n z lim n u n 0 where l is a jacobi operator and the nonlinearities g n s are asymptotically linear as s in the two different cases is a spectral endpoint of l or it belongs to a finite spectral gap of l we obtain the existence of nontrivial solitons of this equation by using variational methods in particular a necessary and sufficient condition is obtained for the existence of gap solitons of the nonlinear equation here solitons appear when we look for standing waves of some discrete nonlinear schrodinger equations

properties of one dimensional anharmonic lattice solitons abstract the existence of bell and kink shaped solitons moving at constant velocity while keeping a permanent profile is studied in infinite periodic monoatomic chains of arbitrary anharmonicity by taking advantage of the equation of motion being integrable with respect to solitons a second order non linear differential equation involving advanced and retarded terms must be solved which is done by implementing a scheme based on the finite element and newton s methods if the potential has a harmonic limit the asymptotic time decay behaves exponentially and there is a dispersion relation between propagation velocity and decay time inversely if the potential has no harmonic limit the asymptotic regime shows up either as a power law or faster than exponential excellent agreement is achieved with toda s model illustrative examples are also given for the fermi pasta ulam and sine gordon potentials owing to integrability an effective one body potential is worked out in each case lattice and continuum solitons differ markedly from one another as regards the amplitude versus propagation velocity relationship and the asymptotic time behavior the relevance of the linear stability analysis when applied to solitons propagating in an infinite crystal is questioned the reasons preventing solitons from arising in a diatomic lattice are discussed

lower and upper estimates on the excitation threshold for breathers in discrete nonlinear schrodinger lattices we propose analytical lower and upper estimates on the excitation threshold for breathers in the form of spatially localized and time periodic solutions in discrete nonlinear schrodinger dnls lattices with power nonlinearity the estimation depending explicitly on the lattice parameters is derived by a combination of a comparison argument on appropriate lower bounds depending on the frequency of each solution with a simple and justified heuristic argument the numerical studies verify that the analytical estimates can be of particular usefulness as a simple analytical detection of the activation energy for breathers in dnls lattices

discrete breathers on symmetry determined invariant manifolds for scalar models on the plane square lattice a group theoretical approach for studying localized periodic and quasiperiodic vibrations in two and three dimensional lattice dynamical models is developed this approach is demonstrated for the scalar models on the plane square lattice the symmetry determined invariant manifolds admitting existence of localized vibrations are found and some types of discrete breathers are constructed on these manifolds a general method using the apparatus of matrix representations of symmetry groups to simplify the standard linear stability analysis is discussed this method allows one to decompose the corresponding system of linear differential equations with time dependent coefficients into a number of independent subsystems whose dimensions are less than the full dimension of the considered system

finite size effects on instabilities of discrete breathers abstract when the two arcs of the continuous phonon spectrum of the floquet matrix of a discrete breather overlap on the unit circle the breather solution in the infinite lattice might be stable while the corresponding solutions in finite systems appear to be unstable more precisely when the model parameters vary the breather in the finite system exhibits a large number of collisions between the floquet eigenvalues belonging to the phonon spectrum these collisions correspond to complex cascades of instability thresholds followed near after by re entrant stability thresholds we interpret this complex structure on the basis of the band analysis of the matrix of the second variation of the action then we can predict that in the limit of an infinite system the number of instability and stability thresholds in the cascade diverges but simultaneously the maximum amplitude of the instabilities vanishes so that the breather in the infinite system recovers its linear stability as long as all its other localized modes remain stable this is the situation which is required in cretegny et al physica d 119 1998 73 87 for having inelastic phonon scattering with two channels we also analyze the size effects when a floquet eigenvalue associated with a localized mode collides with the floquet continuous phonon spectrum with different krein signature in contrast to the previous case the infinite system is unstable after the collision

exact static solutions of a two dimensional discrete ϕ4 model for a two dimensional scalar discrete 4 model we obtain several exact static solutions in the form of the jacobi elliptic functions jef with arbitrary shift along the lattice the quispel roberts thompson type quadratic maps are identified for the considered two dimensional model by using a jef solution we also show that many of the static solutions can be constructed iteratively from these quadratic maps by starting from an admissible initial value the kink solution having the form of tanh is numerically demonstrated to be generically stable

anti integrability in dynamic and variational problems abstract we show using a series of examples that the existence of a singular limit called anti integrable can be useful for proving the existence of chaotic solutions and elucidating some of their properties leading to some new and unexpected results in particular the existence proof of chaotic trajectories in large classes of dynamical systems with discrete time becomes straightforward the method applies even for maps in infinite dimension symplectic or not similar ideas can be used for proving the existence of breathers or self localised modes and of chaotic distributions of breathers in any array of weakly coupled non linear oscillators with continuous time the existence of insulating bipolaronic structures in electron phonon systems electronic self localisation at large electron phonon coupling can also be proven on the same basis

solitons in triangular and honeycomb dynamical lattices with the cubic nonlinearity we study the existence and stability of localized states in the discrete nonlinear schrodinger equation on two dimensional nonsquare lattices the model includes both the nearest neighbor and long range interactions for the fundamental strongly localized soliton the results depend on the coordination number i e on the particular type of lattice the long range interactions additionally destabilize the discrete soliton or make it more stable if the sign of the interaction is respectively the same as or opposite to the sign of the short range interaction we also explore more complicated solutions such as twisted localized modes and solutions carrying multiple topological charge vortices that are specific to the triangular and honeycomb lattices in the cases when such vortices are unstable direct simulations demonstrate that they typically turn into zero vorticity fundamental solitons