bluuebunny/crossref_metadata_2025 huggingface dataset:
Interacting Brownian Particles as a Model of Neural Network
Authors: Iwasaki Ichiro , Nakajima Hiroshi , Shimizu Toshihiro
Published: September 2003

A model of neural network is proposed, in which the dynamics of the internal state of neurons is based on the Brownian motion with the chaotic force, and the bifurcation parameter of the chaotic force is modulated by the position of the Brownian particles. This model is applied to the associative memory problem. The parameter dependence of the dynamics is studied. It is shown that the model can retrieve all embedded patterns and reverse patterns successively. It is found that all neurons exhibit synchronized behavior while a pattern is retrieved.
Characterization of Dolphin Acoustic Echo-Location Data using a Dynamical Classification Method
Authors: Kremliovsky Michael , Kadtke James , Inchiosa Mario , Moore Patrick
Published: September 2003

Many methods for time series analysis derived from nonlinear dynamical systems theory have been developed in the last decade, and have demonstrated remarkable results in a variety of simulated, experimental, and real applications. Classification of time series based on the underlying dynamical generator is also potentially powerful, and we have previously presented a method for dynamical classification based on empirically estimated sets of nonlinear ordinary differential equations, i.e. global dynamical models. A particularly useful area of application for such methods may be biologic and medical data analysis, where few quantitative methods exist for the highly complex time evolutions. Here, as an example of the application of such classification methods, we present an analysis of data of the acoustic pulse trains produced by dolphins as they attempt to echo-locate objects in an ocean environment, which is derived from a controlled experimental framework.
One Method for Restoring Inhomogeneous Attractors
Authors: Janson Natalia B. , Pavlov Alexey N. , Anishchenko Vadim S.
Published: September 2003

Several methods of restoration of phase portraits were applied to real experimental realizations a(t) of biological origin. The algorithms for global reconstruction were used to create qualitative models of the regimes under study. The results of global modeling were satisfactory for the time series of simple shape, but in case of complicated inhomogeneous realizations the traditional algorithms did not give reasonable models. We suggest a method for restoration of inhomogeneous attractors on a(t) as follows: [Formula: see text] while the other coordinates could be restored by any known methods (delay, differentiation, etc.). Such a representation of the attractor's coordinates preserves a simple form of the first equation of the system of differential equations sought [Formula: see text]

This method was tested first on an artificially produced inhomogeneous realization containing pieces with very slow and very quick motion. After that it was successfully applied to real biological inhomogeneous realizations.
Zero-Dispersion Stochastic Resonance in Underdamped SQUIDS
Authors: Golubev G. P. , Kaufman I. Kh. , Luchinsky D. G. , McClintock P. V. E. , Soskin S. M. , Stein N. D.
Published: September 2003

Zero-dispersion stochastic resonance (ZDSR) in underdamped SQUIDs (superconducting quantum interference devices) has been investigated theoretically and by analogue electronic experiment. It is shown that, in common with the standard form of stochastic resonance (SR) that occurs at low frequencies in overdamped bistable systems, ZDSR can result in large noise-induced enhancements of the signal/noise ratio for a weak periodic signal. Unlike standard SR, however, ZDSR is to be anticipated over a vastly wider and tunable range of high frequencies in highly underdamped SQUIDs that need not necessarily be bistable.
Voltage Rectification in a Driven Asymmetric SQUID
Authors: Bartussek R. , Hänggi P. , Sols F. , Zapata I.
Published: September 2003

We argue that the phase across an asymmetric dc SQUID threaded by a magnetic flux can experience a ratchet (periodic and asymmetric) potential. Under an external ac current, a rocking ratchet mechanism operates whereby one sign of the time derivative of the phase is favored. We show that there exists a range of parameters in which a fixed sign of the average voltage across the ring occurs, regardless of the sign of the external current dc component.
Nonclassical Kinetics in Constrained Geometries: Initial Distribution Effects
Authors: Lindenberg K. , Romero A. H. , Sancho J. M.
Published: September 2003

We present a detailed study of the effects of the initial distribution on the kinetic evolution of the irreversible reaction A+B→0 in one dimension. Our analytic as well as numerical work is based on a reaction–diffusion model of this reaction. We focus on the role of initial density fluctuations in the creation of the macroscopic patterns that lead to the well-known kinetic anomalies in this system. In particular, we discuss the role of the long wavelength components of the initial fluctuations in determining the long-time behavior of the system. We note that the frequently studied random initial distribution is but one of a variety of possible distributions leading to interesting anomalous behavior. Our discussion includes an initial distribution with correlated A-B pairs and one in which the initial distribution forms a fractal pattern. The former is an example of a distribution whose long wavelength components are suppressed, while the latter exemplifies one whose long wavelength components are enhanced, relative to those of the random distribution.
Noise Induced Order: Stochastic Resonance
Authors: Schimansky-Geier Lutz , Freund Jan A. , Neiman Alexander B. , Shulgin Boris
Published: September 2003

We investigate stochastic resonance in the framework of information theory. Input signals are taken from an electronic circuit and output signals are produced by a Schmitt trigger. These electronic signals are analyzed with respect to their informational contents. Conditional entropies and Kullback measures exhibit extrema for values of noise intensity in the range of stochastic resonance. However, it has to be noted that these extrema are related to synchronization effects, observed in stochastic resonance for large signal amplitudes, rather than to a peak in the related spectrum indicating some periodic component.
Dynamics of Globally Coupled Rotators with Multiplicative Noise
Authors: Kim Seunghwan , Park Seon Hee , Ryu Chang Soo , Han Seung Kee
Published: September 2003

We study the dynamics of globally coupled rotator model with multiplicative noise concentrating on time-delay effect. It is shown that at a critical noise intensity the system undergoes a noise-induced transition and is split into clusters. Time-delayed interaction in the system affects the picture of the transition suppressing the clustering of the rotators and generates the switching phenomenon between the clusters. For some values of delay time the system has two stable steady states which show different dynamics. According to the initial condition the system evolves into one of the stable steady states. We discuss the nature of the transition and the switching phenomenon in the system.
Stochastic Cluster Dynamics of Macromolecules
Authors: Chikishev Andrej Yu. , Netrebko Nina V. , Romanovsky Youri M. , Ebeling Werner , Schimansky-Geier Lutz , Netrebko Aleksej V.
Published: September 2003

We study the motion of a particle in a two-dimensional potential landscape containing potential minima under influence of random and parametric forcing. The influence of the finite size of the particles is investigated. Possible connections to proton transitions and ligand penetrations in biomolecules are discussed.
A Stochastic Model of Polymer Translocation Dynamics Through Biomembranes
Authors: Park P. J. , Sung W.
Published: September 2003

The polymer translocation across membranes is studied as a stochastic process over a free energy barrier, where the mean first passage time is examined analytically as a measure of translocation rate. The nonequilibrium fluctuations of chemical potential difference enhance directed transport of polymers, giving rise to a resonant activation. The translocation time of a long-chain polymer is found to be very sensitive not only to the average value of chemical potential difference but also to its fluctuation.
Spatio-Temporal Dynamics in Semiconductor Lasers with Delayed Optical Feedback
Authors: Münkel Markus , Kaiser Friedemann , Hess Ortwin
Published: September 2003

We investigate the influence of delayed optical feedback (DOF) on the dynamics of semiconductor lasers. In the case of the narrow single-stripe laser, we find that the presence of DOF leads to a wealth of dynamical phenomena in the coherence-collapsed regime, including mode-hopping between compound-cavity modes induced by DOF. Focusing on the twin-stripe laser — the most simple system with inherent spatio-temporal instabilities — we show that feedback may both induce and suppress spatio-temporal instabilities. Eigenmode analysis enables us to determine and identify the underlying spatio-temporal "supermodes". For appropriately chosen parameters, regular regimes including continuous wave operation can be obtained from an originally chaotic regime. For moderate to strong feedback, interaction between the spatial degrees of freedom in the twin-stripe laser and the compound cavity modes leads to a new phenomenon which we term "spatio-temporal mode-hopping".
Spatial and Temporal Self-Organization of Bacterial Population
Authors: Bagayev S. N. , Gusev V. A. , Orlov V. A. , Panov S. V.
Published: September 2003

The peculiarities of self-organization of a bacterial population are analyzed. Comparison of the dynamic characteristics of a bacterial population received experimentally by biological and optical methods is made. It is shown that the observable dynamic properties of population are connected with its spatial self-organization. Such behavior of moving microorganisms is a consequence of the search for optimum conditions of viability retention.
Nonlinear Dynamics of a Model of Interacting Waves in a Space with Partially Reflecting Boundaries
Authors: Chetverikov Alexandr P.
Published: September 2003

We investigate numerically the spatio-temporal behavior of a bounded distributed nonlinear dynamical system of finite length that describes the interaction of two one-dimensional waves. The nonlinear and dispersion properties of the waves and conditions at the boundaries of the interaction space correspond to a simple case, when the electromagnetic wave in a waveguide interacts with the wave in the flow of oscillating electrons. Typical spatial structures of the waves, their temporal behavior and bifurcational transitions in the parametric plane are examined on the basis of the qualitative analysis and statistical processing of the data of the computer experiments.
The Explicit Solutions of Frobenius-Perron Equation for the Chaotic Infinite Maps
Authors: Golubentsev A. F. , Anikin V. M.
Published: September 2003

The simple one-dimensional mappings which demonstrate the chaotic behavior on the infinite interval and the appertaining explicit invariant densities having the forms of Cauchy distribution, F-distribution, and Z-distribution are presented.
Patchiness: A New Diagnostic for Lagrangian Trajectory Analysis in Time-Dependent Fluid Flows
Authors: Malhotra Naresh , Mezić Igor , Wiggins Stephen
Published: September 2003

In 2-D time-dependent fluid flows, a patch represents a localized region in space that has a significantly different average velocity compared to its surroundings. We show that one can obtain important information about the Lagrangian particle motion in such flows by studying the nature, long-term evolution, and statistical characteristics of the patchiness behavior. For example, the dispersion of passive tracers at any time is directly related to the distribution of patches in the flow. We thoroughly investigate the transport properties of the Lagrangian trajectories associated with a cellular flow previously used as a model for time-dependent Rayleigh–Bénard convection, and a kinematic model of a meandering jet (originally due to Bower [1991]). In both cases, we examine the statistical attributes of the patchiness, their relationship with the geometric features of the stable and unstable manifolds, and the effect of noise on the structure of patchiness. We uncover some interesting features associated with the origin of these patches and their influence on Lagrangian transport.
Cantor Staircases in Physics and Diophantine Approximations
Authors: Losada M. Piacquadio , Grynberg S.
Published: September 2003

For a wide class of dynamical systems the variables involved relate to one another through a Cantor staircase function. When they are time variables, the staircases have well-known universal properties that suggest a connection with certain classical problems in Number Theory. In this paper we extend some of those universal properties to certain Cantor staircases that appear in Quantum Mechanics, where the variables involved are not time variables. We also develop some connections between the geometry of these Cantor staircases and the problem of approximating irrational numbers of rational ones, classical in Number Theory.
Spiral Waves are Stable in Discrete Element Models of Two-Dimensional Homogeneous Excitable Media
Authors: Feldman Andrew B. , Chernyak Yuri B. , Cohen Richard J.
Published: September 2003

The spontaneous breakup of a single spiral wave of excitation into a turbulent wave pattern has been observed in both discrete element models and continuous reaction–diffusion models of spatially homogeneous 2D excitable media. These results have attracted considerable interest, since spiral breakup is thought to be an important mechanism of transition from the heart rhythm disturbance ventricular tachycardia to the fatal arrhythmia ventricular fibrillation. It is not known whether this process can occur in the absence of disease-induced spatial heterogeneity of the electrical properties of the ventricular tissue. Candidate mechanisms for spiral breakup in uniform 2D media have emerged, but the physical validity of the mechanisms and their applicability to myocardium require further scrutiny. In this letter, we examine the computer simulation results obtained in two discrete element models and show that the instability of each spiral is an artifact resulting from an unphysical dependence of wave speed on wave front curvature in the medium. We conclude that spiral breakup does not occur in these two models at the specified parameter values and that great care must be exercised in the representation of a continuous excitable medium via discrete elements.
The Hénon and Heiles Problem in Three Dimensions.
Authors: Ferrer S. , Lara M. , Palacián J. , Juan J. F. San , Viartola A. , Yanguas P.
Published: September 2003

This paper is the first part of a study of the Hénon and Heiles problem in three dimensions. Due to the axial symmetry of the Hamiltonian, the third component of the angular momentum is an integral and the system is considered as a Hamiltonian with two degrees of freedom. As functions of that integral, we show the existence of three circular trajectories around the axis Oz and a domain for which we have bounded motions. In the part of that domain near the origin, the corresponding dynamical system is treated as a perturbed harmonic oscillator in 1–1–1 resonance. We present some numerical studies searching for periodic orbits, showing the corresponding Poincaré surfaces of section. In addition, we obtain some natural families of periodic orbits associated with the relative equilibria of the fourth order normalized system.
Detecting Nonlinearity in Experimental Data
Authors: Small Michael , Judd Kevin
Published: September 2003

The technique of surrogate data has been used as a method to test for membership of particular classes of linear systems. We suggest an obvious extension of this to classes of nonlinear parametric models and demonstrate our methods with respiratory data from sleeping human infants. Although our data are clearly distinct from the different classes of linear systems we are unable to distinguish between our data and surrogates generated by nonlinear models. Hence we conclude that human respiration is likely to be a nonlinear system with more than two degrees of freedom with a limit cycle that is driven by high dimensional dynamics or noise.
Chaotic Wakes and Other Wave-Induced Behavior in a System of Reaction–Diffusion Equations
Authors: Davidson F. A.
Published: September 2003

The phenomenon of spatiotemporal chaos as it arises in systems of reaction–diffusion equations has been the subject of much study over recent years. Initially, chaotic responses in these systems were found to occur when the ratio of the diffusion coefficients was chosen sufficiently large (greater than one). Subsequently, such behavior has also been found when this ratio is taken to be unity with the conclusion that this represents a new mechanism for inducing chaos. In this paper we discuss numerical simulations of a system of reaction–diffusion equations which has been derived to study the large-scale growth patterns of fungal colonies. We show that, for this reaction–diffusion system at least, these apparently different mechanisms represent two ways of entering the same "chaotic region" in parameter space. We also discuss other wave-induced behavior which characterize the routes to chaos.
Unpinning of a Vortex in a Chemical Excitable Medium
Authors: Huyet G. , Dupont C. , Corriol T. , Krinsky V.
Published: September 2003

Rotating vortices are easily pinned to any defects in excitable media. Moderate advective fields cannot remove the pinned vortices. We have found and tested experimentally a mechanism for unpinning a vortex. To unpin a vortex, we created additional waves close to its core. A vortex can be unpinned when a new created wave is pinned to the same obstacle and has a topological defect. This mechanism works perfectly both in experiment and in numerical simulations. A hypothesis is put forward that similar mechanisms may underlay pacing for tachycardia control used in cardiac clinics.
Complex Dynamic Behavior in a Model of Electro-Optic Device with Time-Delayed Feedback
Authors: Shobukhov Andrew V.
Published: September 2003

Oscillatory and chaotic regimes arising from Andronov-Hopf bifurcation are obtained in a model of fiber-optic interferometric strain gauge with delayed nonlinear feedback as a result of analytical and numerical investigation. Complicated behavior of solutions in the case of infinite number of equilibrium points is reported.
Differential-Algebraic Equations: A Tutorial Review
Authors: Beardmore R. E. , Song Y. H.
Published: September 2003

This article (Funded by EPSRC and the National Grid Company.) explores some introductory principles of differential-algebraic equations (DAEs) and makes a connection with the theory of dynamical systems. Some results which are new in the field of DAEs are also surveyed.

Most treatments on DAE emphasize the differences that exist when compared with the ODE case. Here we seek to underline the similarities so that readers with a very basic knowledge of nonlinear dynamics can understand some of their consequences in this more general context.
Experimental Analysis of Codimension-2 Bifurcations in a Periodically-Forced Opto-Thermal Oscillator
Authors: Farjas J. , Herrero R. , Pi F. , Orriols G.
Published: September 2003

The response of a self-oscillating opto-thermal device irradiated with modulated light is investigated both numerically and experimentally. The subharmonic bifurcation structure of the amplitude-frequency parameter plane is analyzed and a variety of local and global codimension-2 bifurcations are characterized. A detailed study of the dynamics transformation around these points is also provided.
Lie Algebras, Structure of Nonlinear Systems and Chaotic Motion
Authors: Banks S. P. , McCaffrey D.
Published: September 2003

The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced, based on the structure theory of Lie algebras.
Routes to Chaos in Neural Networks with Random Weights
Authors: Albers D. J. , Sprott J. C. , Dechert W. D.
Published: September 2003

Neural networks are dense in the space of dynamical system. We present a Monte Carlo study of the dynamic properties along the route to chaos over random dynamical system function space by randomly sampling the neural network function space. Our results show that as the dimension of the system (the number of dynamical variables) is increased, the probability of chaos approaches unity. We present theoretical and numerical results which show that as the dimension is increased, the quasiperiodic route to chaos is the dominant route. We also qualitatively analyze the dynamics along the route.
Asymptotic Independence and Uniform Distribution of Quantization Errors for Spatially Discretized Dynamical Systems
Authors: Diamond P. , Vladimirov I.
Published: September 2003

Computer simulation of dynamical systems involves a state space which is the finite set of computer arithmetic. Restricting state values to this grid produces roundoff effects which can be studied by replacing the original system with a spatially discretized dynamical system. Study of the deviation of the discretized trajectories from those of the original system reduces to that of appropriately defined quantization errors. As the grid is refined, the asymptotic behavior of these quantization errors follows probabilistic laws. These results are applied to discretized polynomial mappings of the unit interval.
Synchronization of Chua's Oscillators with the Third State as the Driving Signal
Authors: Wang Xiao Fan , Wang Zhi Quan
Published: September 2003

In this paper, we study the use of the third state as the driving signal to achiee synchronizaton of two Chua's oscillators. We prove that globally asymptotic synchronization can be attained via linear error feedback. Simulation results are presented to verify the operation of the design.
Numerical Detection and Continuation of Homoclinic Points and Their Bifurcations for Maps and Periodically Forced Systems
Authors: Yagasaki K.
Published: September 2003

This paper describes a driver HomMap to the standard local bifurcation software AUTO for numerical analysis of homoclinic and heteroclinic bifurcations in maps and periodically forced systems. The driver can detect loci of homoclinic points if the unstable or stable manifolds are of dimension one, and treat problems of two or more dimensions. The algorithms used and their implementations in AUTO with HomMap are explained. Three examples are given, namely the Hénon map and the single and coupled Duffing oscillators, in order to demonstrate the usefulness and capabilities of the driver.
Pretzel Orbits in Simulations of Epitaxial Crystal Growth
Authors: Menke Christoph
Published: September 2003

A third order autonomous ordinary differential equation is studied that describes stationary solutions of a nonlinear partial differential equation. The PDE models the growth of an epitaxial film on misoriented crystal substrates and is similar to the Kuramoto–Sivashinsky equation, but contains an additional nonlinear term. The equilibria, the periodic solutions, and the heteroclinic orbits of the ODE are analyzed, and stability results are given. Parameter regions are identified where the equilibria and the periodic solutions are unstable, but other bounded solutions exist. Their phase portrait is a double focus ("pretzel") that connects the stable and the unstable manifolds of the equilibria.
The Control of Chaos: Theoretical Schemes and Experimental Realizations
Authors: Arecchi F. T. , Boccaletti S. , Ciofini M. , Meucci R. , Grebogi C.
Published: September 2003

Controlling chaos is a process wherein an unstable periodic orbit embedded in a chaotic attractor is stabilized by means of tiny perturbations of the system. These perturbations imply goal oriented feedback techniques which act either on the state variables of the system or on the control parameters. We review some theoretical schemes and experimental implementations for the control of chaos.
Adaptive Mode Selection Using On–Off Switching of Chaos
Authors: Davis Peter
Published: September 2003

Onset of chaotic mode transition can be used in a simple and robust mechanism for adaptation in multimode systems. We present an idealized model of adaptive mode selection using on–off switching of chaotic basin transitions in a multistable system. A stochastic description obtained in the limit of large switching intervals gives search times in terms of projections of the chaotic state onto basins of multistability.
Adaptive Mode Selection Based on Chaotic Search in a Fabry–Perot Laser Diode
Authors: Liu Yun , Davis Peter
Published: September 2003

This paper describes mode-transition dynamics of a compound-cavity, multimode, Fabry–Perot (FP) semiconductor laser and the basic idea of adaptive wavelength selection using chaotic search. In the presence of external feedback, the laser is shown to exhibit different mode-competition dynamics including single-mode, multistability, and chaotic mode-transition states. A chaotic search technique is proposed to adaptively select a dominant lasing mode which fits the environment by switching a control parameter (feedback level or bias injection) between multistability and chaotic mode-transition states. The effectiveness of the adaptive mode selection is verified through a computer simulation and the robustness as well as the experimental implementation of the technique are discussed.
Dynamic Stabilization of Unstable Periodic Orbits in a CO2 Laser by Slow Modulation of Cavity Detuning
Authors: Pisarchik A. N. , Corbalán R. , Chizhevsky V. N. , Vilaseca R. , Kuntsevich B. F.
Published: September 2003

We demonstrate numerically and experimentally that a slow modulation of cavity detuning in a loss-modulated CO 2 laser can stabilize unstable periodic orbits even when the system remains in a particular dynamical regime for adiabatic changes of the detuning. When the parameter changes faster than the transient response of deformation of the original periodic attractor, the system can evolve toward an unstable periodic orbit.
Chaos Control in External Cavity Laser Diodes using Electronic Impulsive Delayed Feedback
Authors: Naumenko A. V. , Loiko N. A. , Turovets S. I. , Spencer P. S. , Shore K. A.
Published: September 2003

In this paper an examination is made of the performance characteristics of a scheme of chaos control using optimised impulsive delayed feedback. The scheme is applied in a model of external cavity laser diodes giving attention to the application of the feedback via modulation of the laser drive current taking into account practical constraints arising from technical delay and the frequency in application of the control signal. It is demonstrated that by use of a so-called ∊-ball condition chaos control is achievable in the fully developed coherence collapse regime without preliminary targeting dynamics to an unstable orbit.
Nonlinearity Tests Using the Extrema of a Time Series
Authors: di Garbo A. , Balocchi R. , Chillemi S.
Published: September 2003

The analytical properties of the solution of a system of ODEs in the complex time plane influence its dynamical behavior on the real time axis. In particular, the extrema of the real time solution can be associated to the singularities of the complex solution falling close to the real time axis. Moreover for a twice differentiable stochastic process, the expected value of the number of extrema for unit time can be determined. These two results are used here as the starting point to introduce two new algorithms to test for time series nonlinearity. They do not require the phase space reconstruction protocol and seem to work well also for short data sets.
Transmission of Signals via Synchronization of Chaotic Time-Delay Systems
Authors: Pyragas K.
Published: September 2003

Secure communication via chaotic synchronization is demonstrated using dynamical systems governed by delay deferential equations. Strange attractors of such systems can have an arbitrarily large number of positive Lyapunov exponents giving rise to very complex time signals. This features can provide high security of masked messages.
Diffusion Parameter Control of Spatiotemporal Chaos
Authors: Montagne Raul , Colet Pere
Published: September 2003

The stabilization of periodic solutions in the regime of spatiotemporal chaos through a diffusion parameter control is studied. We show that unstable plane waves in the Complex Ginzburg–Landau equation can be effectively stabilized in choatic regimes such as phase turbulence and spatiotemporal intermittency or defect turbulence.
Hypernormal Form for the Hopf-Zero Bifurcation
Authors: Algaba A. , Freire E. , Gamero E.
Published: September 2003

In this paper we present hypernormal forms up to an arbitrary order for equilibria of tridimensional systems having a linear degneracy corresponding to a pair of pure imaginary eigenvalues and a third one zero. These simplest normal forms are obtained assuming some generic conditions on the quadratic terms, and using C∞-conjugacy as well as C∞-equivalence. Also, the case of ℤ2-symmetric systems is considered. In this situation, the hypernormal forms are characterized under generic conditions on the cubic terms. In all the cases, we provide recursive algorithms that compute explicitly the hypernormal form coefficients, in terms of the normal form coefficients.
Bifurcation Analysis of Periodic Solutions of Neural Functional Differential Equations: A Case Study
Authors: Engelborghs Koen , Roose Dirk , Luzyanina Tatyana
Published: September 2003

This paper deals with the numerical bifurction analysis of periodic solutions of a system of neutral functional differential equations (NFDEs). Compared with retarded functional differential equations, the solution operator of a system of NFDEs does not smooth the initial data as time increases and it is no longer a compact operator. The stability of a periodic solution is determined both by the point spectrum and by the essential spectrum of the Poincaré operator. We show that a periodic solution can change its stability not only by means of a "normal" bifurcation but also when the essential spectrum crosses the unit circle. In order to monitor the essential spectrum during continuation, we derive an upper bound on its spectral radius. The upper bound remains valid even at points where the radius of the essetial spectrum is noncontinuous. This can occur when the delay and the period are rationally dependent. Our numerical results present these new dynamical phenomena and we state a number of open questions. Although we restrict our discussion to a specific example, we strongly believe that the issues we discuss are representative for a general class of NFDEs.
Perturbations of 8-Attractors and Births of Satellite Systems
Authors: Goetz Arek
Published: September 2003

We investigate the most basic two-dimensional generalizations of interval exchange maps. We show that a recently discovered attracting piecewise rotation system with two atoms becomes repelling under a significant perturbation of one of the centers of rotation. However, small perturbations of this system result in remarkable bifurcations into local attracting systems and they give birth to new satellite periodic cells. Bifurcations can be repeated recursively and eventually lead to piecewise rotations with infinitely many satellite periodic cells.
Bounded Geometry in the Supports of Ergodic Invariant Probability Measures
Authors: Hu Jun
Published: September 2003

In this paper we put some techniques and methods of [Misurewicz, 1979; Hu & Sullivan, 1997; Hu & Tresser, 1998; Blokh & Lyubich, 1990; Martens et al., 1992] together to show that if a map f, from an interval I into itself with finitely many turning points, satisfies a new smooth regularity in [Hu & Sullivan, 1997] and is on the boundary of chaos, and if μ is an ergodic f-invariant probability measure on I which is not concentrated on a periodic orbit of f, then the support K of μ is a Cantor set of bounded geometry, and hence has Lebesgue measure 0 and Hausdorff dimension strictly between 0 and 1. We also include some natural examples which satisfy this new smooth regularity rather than the traditional ones.
A Model of Two Nonlinear Coupled Oscillators for the Study of Heartbeat Dynamics
Authors: di Bernardo Diego , Signorini Maria G.
Published: September 2003

The cardiac conduction system may be assumed to be a network of self-excitatory pacemakers, with the SinoAtrial (SA) node having the highest intrinsic rate. Subsidiary pacemakers with slower firing frequencies are located in the AtrioVentricular (AV) node and the His-Purkinje system. Under physiological conditions, the SA node is the dominant pace-maker and impulses travel from this node to the ventricle through the AV junction, which is traditionally regarded as a passive conduit. We consider the Av node as an active pace-maker and develop a model of two nonlinear coupled oscillators in order to describe the interaction between the SA and the AV node. These two nonlinear oscillators are based on a modification of the van der Pol osciallator, so that the generated waveforms resemble the action potentials of cells in the SA and the AV node respectively. A bifurcation analysis of this model is performed and the pathophysiological different kinds of heartbeat pathologies (1°, 2° (both Wenckebach and non-Wenckebach) and 3° AV blocks, sinus arrest, atrials bigeminy, etc.). This simple nonlinear model helps to improve the understanding of the complex phenomena involved in heart rhythm generation as well as of heart rate control and function.
CNN Genes for One-Dimensional Cellular Automata: A Multi-Nested Piecewise-Linear Approach
Authors: Dogaru Radu , Chua Leon O.
Published: September 2003

This paper introduces a novel CNN cell which guarantees the implementation of any local rule on three variables defined by a Boolean truth table. Moreover, since the output of the cell is completely specified by a simple mathematical formula, it is possible to develop a systematic theory for locating those regions in the CNN genes parameter space where complex behaviors may occur. The output cell formula is a simple piecewise-linear function, and for the case of a one-dimensional CNN the entire set of 256 CNN genes associated with the corresponding local Boolean functions are listed in this paper.
On the Formation and Breakup of Spiral Waves of Calcium
Authors: Mckenzie A. , Sneyd J.
Published: September 2003

We study spiral waves in a model of Ca2+dynamics in the Xenopus laevis oocyte. Spiral waves in the model were initiated by simulating the release of inositol 1,4,5-trisphosphate ( IP3), a common experimental protocol. No artificial heterogeneities needed to be imposed on the system for the spontaneous formation of spiral waves. Increasing the size of the IP3additions caused a decrease in the rotation period, and the breakup of the spiral wave, irregular spatiotemporal patterns occurred. Similar disorganized patterns are sometimes seen experimentally; based on our simulations we predict that a simple experimental procedure may be sufficient to reproduce unstable spiral waves.
Four Indeterminate Pitchfork Bifurcations Resulting in Unpredictable Dynamics in Nonlinear Driven Oscillators
Authors: Soliman Mohamed S.
Published: September 2003

Subcritical pitchfork bifurcations occur when the equilibrium solution loses its stability as it intersects a secondary unstable path. In this paper we outline the phenomenon of indeterminate subcritical pitchfork bifurcatons. These arise when the equilibrium solution is located on a fractal boundary as it loses its stability at the bifurcation. The outcome is then unpredictable even under infinitesimal perturbations from the equilibrium solution.
Adaptive Control of Chaotic Systems with Uncertainties
Authors: Zhang Huaizhou , Qin Huashu , Chen Guanrong
Published: September 2003

In this paper, an adaptive control scheme, that employs a Gaussian radial basis function network with output weights updated on-line according to the Lyapunov stability theory, is suggested for regulation of a class of chaotic systems with uncertainties. Theoretical analysis guarantees that under the control of the proposed adaptation law, uncertain chaotic systems can asymptoticaly track target orbits within arbitrarily small tolerance bounds. As an example, control of the uncertain Duffing–Holmes system is presented with computer simulations, which verifies and visualizes the theory and design of the adaptive controller.
Error Performance of Chaotic Digital Code-Divison Multiple Access (CDMA) Systems
Authors: Yang Tao , Chua Leon O.
Published: September 2003

Chaotic digital code-division multiple access ((CD)2MA) systems are new types of communication systems using chaotic carriers. (CD)2MA had found applications in both wireless and other communication systems. In this paper we study the error performance of chaotic digical code-division multiple access ((CD)2MA) systems used in both wireless environment and cable networks. We present first the details of the interleave mode of the (CD)2MA. When a (CD)2MA system works in the interleave mode, each of its transmitters only transmits signals during a portion of the bit duration of the message signal. The total interference level is thus reduced which in turn increases the channel capacity. The relationship between the bit error rate (BER) and the interleave rate is studied in both the synchronous (CD)2MA system used in cable systems, and the asynchronous (CD)2MA system used in wireless environments. Our simulation results show that in a cable system the synchronous (CD)2MA system can support 1.5 times more channel capacity than synchronous CDMA systems, and double the channel capacity of CDMA systems in wireless environments.
The Loss of Predictability of Monetary Policy in a Macrodynamic System: Highly Intertwined Absolute and Transient Basins of Attraction
Authors: Soliman A. S.
Published: September 2003

The implications of studying basins of attraction as regards monetary policy in a model describing inflation and unemployment is presented. It is shown that when basin boundaries are highly intertwined this may lead to unpredicability in monetary policy, since we cannot determine the final states of the system. We illustrate how basins of attraction may be used to determine whether successive small reductions in the monetary growth rate (gradualist approach), as opposed to a one time reduction (shock approach), would be more successful in achieving the desired levels of inflation (expected inflation) and unemployment. Furthermore, an example is presented on how the concept of transient basins, could be more relevant in the study of economic systems in the ever-changing environment.
Synchronized Chaos in Coupled Double Disk Homopolar Dynamos
Authors: Moroz Irene M. , Smith Leonard A. , Hide Raymond
Published: September 2003

Symmetrically coupled systems of N self-exciting Faraday disk homopolar dynamos have been proposed by Hide [1997] as a testbed for both the analytical and numerical study of the dynamics of coupled dynamo systems. There arises a natural hierarchy of systems, consisting of a series of N single disk dynamos, each in series with a motor [Hide et al., 1996]. Synchronization in the dynamics of these systems is investigated. The transition from the special case of strict amplitude and linear phase locking to the case of more general synchronization is examined. The particular case of two magnetically coupled homopolar dynamos with dissimilar characteristics is considered; the behavior of this N = 2 case is governed by six-coupled nonlinear ordinary differential equations which contain a total of thirteen dimensionless parameters. It is proved that for sufficiently small perturbations, the states of each of the two component dynamos are locked together. Numerical results suggest that this locking extends to finite perturbations, the state of one system being a linear function of the state of the other. As the size of the perturbation increases, this strict phase locking is lost, yet the complicated chaotic trajectories of each dynamo appear to remain strictly synchronized. This paper is therefore the result of a previous study of coupled dynamos [Moroz et al., 1998] in which no more than five of the parameters were assumed to be independent.
Basin Properties in Two-Dimensional Noninvertible Maps
Authors: Cathala J. C.
Published: September 2003

Properties of the basins of noninvertible maps of the plane are studied using the method of critical curves. The paper considers the class of maps having a region of the phase plane where the number of first rank preimages is greater than two. More particularly, the paper gives the conditions of existence of a connected basin, a nonconnected basin and a multiply connected basin in maps where the segments of the critical curve separate the plane into regions, one with three first rank preimages and the other with only one preimage. Then, maps having a maximum number of first rank preimages equal to four are considered.
Wavelet Reconstruction of Nonlinear Dynamics
Authors: Allingham David , West Matthew , Mees Alistair I.
Published: September 2003

We investigate the reconstruction of embedded time-series from chaotic dynamical systems using wavelets. The standard wavelet transforms are not applicable because of the embedding, and we use a basis pursuit method which on its own does not perform very well. When this is combined with a continuous optimizer, however, we obtain very good models. We discuss the success of this method and apply it to some data from a vibrating string experiment.
An Algorithm for Estimating Fixed Points of Dynamical Systems from Time Series
Authors: Aguirre Luis A. , Souza Álvaro V. P.
Published: September 2003

This paper presents an algorithm for estimating fixed points of dynamical systems from time series. In some cases the new procedure can accurately estimate fixed points of which there is very little information in the data. Another advantage is that, although no prior knowledge is assumed, the new algorithm permits the user to employ a priori information about the system such as symmetry and the existence of a trivial fixed point. The new algorithm is tested on the Lorenz and Rössler systems and on real data taken from Chua's circuit.
A Note on Synchronization with Heterogeneous Driving
Authors: Wang Xinmin , Tang Binyong , Ruan Jiong
Published: September 2003

Synchronization in chaotic systems with heterogeneous driving is investigated in this letter, and is shown to be impossible in general.
Observer-Based Robust Synchronization of Dynamical Systems
Authors: Pogromsky Alexander , Nijmeijer Henk
Published: September 2003

The paper deals with the problem of robust synchronization of dynamical systems. The design procedure is based on the concept of observers with absolutely stable error dynamics. In the general case of nonlinear time-varying error dynamics the procedure requires exact knowledge of a Lyapunov function while in case of the linearizable error dynamics frequency domain conditions which ensure existence of such a function can be employed. Two examples are considered: synchronization of two Lorenz systems and Rössler systems.
Experimental Evidence of a Hard Transition to Chaos
Authors: Río E. Del , Sanmartín J. R. , López-Rebollal O.
Published: September 2003

A generic, sudden transition to chaos has been experimentally verified using electronic circuits. The particular system studied involves the near resonance of two coupled oscillators at 2 : 1 frequency ratio when the damping of the first oscillator becomes negative. We identified in the experiment all types of orbits described by theory. We also found that a theoretical, 1D limit map fits closely a map of the experimental attractor which, however, could be strongly disturbed by noise. In particular, we found noisy periodic orbits, in good agreement with noise theory.
Computation of Normal Forms of Differential Equations Associated with Non-Semisimple Zero Eigenvalues
Authors: Bi Q. , Yu P.
Published: September 2003

This paper presents a method to compute the normal forms of differential equations whose Jacobian evaluated at an equilibrium includes a double zero or a triple zero eigenvalue. The method combines normal form theory with center manifold theory to deal with a general n-dimensional system. Explicit formulas are derived and symbolic computer programs have been developed using a symbolic computation language Maple. This enables one to easily compute normal forms and nonlinear transformations up to any order for a given specific problem. The programs can be conveniently executed on a main frame, workstation or a PC machine without any interaction. Mathematical and practical examples are presented to show the applicability of the method.
Edge of Chaos and Local Activity Domain of the Gierer–Meinhardt CNN
Authors: Dogaru Radu , Chua Leon O.
Published: September 2003

This paper present an application of the local activity theory [Chua, 1998] to a specific reaction–diffusion cellular nonlinear network (CNN) with cells defined by the model of morphogenesis first proposed in [Gierer & Meinhardt, 1972]. Both the local activity domain and a subset called the "edge of chaos" are identified in the cell parameter space. Within these domains, various cell parameter points were selected to illustrate the effectiveness of the local activity theory in choosing the parameters for the emergence of complex (static and dynamic) patterns in a homogeneous lattice formed by coupled locally active cells.
Higher-Order Spectra of Nonlinear Polynomial Models for Chua's Circuit
Authors: Elgar Steve , Vanhoff Barry , Aguirrre Luis A. , Freitas Ubiratan S. , Chandran Vinod
Published: September 2003

Polynomial models are shown to simulate accurately the quadratic and cubic nonlinear interactions (e.g. higher-order spectra) of time series of voltages measured in Chua's circuit. For circuit parameters resulting in a spiral attractor, bispectra and trispectra of the polynomial model are similar to those from the measured time series, suggesting that the individual interactions between triads and quartets of Fourier components that govern the process dynamics are modeled accurately. For parameters that produce the double-scroll attractor, both measured and modeled time series have small bispectra, but nonzero trispectra, consistent with higher-than-second order nonlinearities dominating the chaos.
Control of Chaos Using Sampled-Data Feedback Control
Authors: Yang Tao
Published: September 2003

In this paper we present a theory for control of chaotic systems using sampled data. The output of the chaotic system is sampled at a given sampling rate and the sampled output is used by a feedback subsystem to construct a control signal, which is held constant by a holding subsystem. Hence, during each control iteration, the control input remains unchanged. Theoretical results on the asymptotic stability of the resulting controlled chaotic systems are presented. Numerical experimental results via Chua's circuit are used to verify the theoretical results.
Symmetry Breaking of Periodic Orbits in Control Systems: A Harmonic Balance Approach
Authors: Aguirre Baltazar , Alvarez-Ramírez Jose , Suárez Rodolfo
Published: September 2003

This work is concerned with linear control systems subjected to saturated feedback. A first harmonic approach is used to describe the existence of nonsymmetric periodic orbits in a three-dimensional control system. By taking a high-gain parametrization of the feedback control, the presence of nonsymmetric (first harmonic) periodic orbits is demonstrated for certain values of the parameter. Since it is also shown that nonsymmetric periodic orbits do not exist for small and large values of the parameter, evidences are found of the existence of symmetry breaking bifurcations.