bluuebunny/crossref_metadata_2025 huggingface dataset:
On the Synchronization of Chaos Systems by Using State Observers
Authors: Morgül Ömer , Solak Ercan
Published: September 2003

We show that the synchronization of chaotic systems can be achieved by using the observer design techniques which are widely used in the control of dynamical systems. We prove that local synchronization is possible under relatively mild conditions and global synchronization is possible if the chaotic system has some special structures, or can be transformed into some special forms. We show that some existing synchronization schemes for chaotic systems are related to the proposed observer-based synchronization scheme. We prove that the proposed scheme is robust with respect to noise and parameter mismatch under some mild conditions. We also give some examples including the Lorenz and Rössler systems and Chua's oscillator which are known to exhibit chaotic behavior, and show that in these systems synchronization by using observers is possible.
An Experimental Verification of Basin Metamorphoses in A Nonlinear Mechanical System
Authors: Todd M. D. , Virgin L. N.
Published: September 2003

This paper describes bifurcations and the basin boundary metamorphoses that give rise to post-fold outcome indeterminacy from a primarily experimental perspective. A gravity-loaded cart-and-track system is constrained to mimic the twin-well, single-degree-of-freedom Duffing oscillator. Of primary interest is the study of how motion, initially contained within a single well, "spills over" into the adjacent well. Although this system is globally bounded, it retains the same generic features of the single-well canonical escape equation. Using time-embedded coordinates, the technique of stochastic interrogation is used to generate the initial condition maps at three different forcing levels corresponding to three different regimes of post-fold outcomes. These three regions are characterized, respectively, by smooth basin boundaries with safe jumps to resonance, fractal basin boundaries with jumps that may or may not restabilize on to the resonant attractor, and eroded basins with unsafe jumps leading to escape from the local well. This experiment successfully replicates much of the subtle global behavior observed in numerical simulations.
Dispersion Relation of Spreading Depression Waves in the Chicken Retina
Authors: Brand S. , Dahlem M. A. , de Lima V. M. Fernades , Müller S. C. , Hanke W.
Published: September 2003

To describe the spatio-temporal patterns of excitable media, the relationship between frequency and propagating velocity of a wave (dispersion relation) is an important issue. Spreading depression (SD) waves in the brain and especially in the retina are an example of self-organized waves in excitable media (here in neuronal tissue). In the retina such waves are accompanied by a remarkable intrinsic optic signal (IOS), which allows to study the wave by standard video imaging techniques. In this study the dispersion relation of SD waves was investigated under different conditions. As expected from standard theories of excitable media, the velocity is a monotone function of the repetition rate, i.e. of the recovery state, with an absolute refractory phase of about 2.4 min. By increasing the temperature, the shape of the curve changes in a significant way. Physiological and theoretical considerations are given for the variation of the dispersion relation.
Simulating Quadratically Nonlinear Random Processes
Authors: Vanhoff Barry , Elgar Steve
Published: September 2003

A technique to generate realizations of quadratically nonlinear non-Gaussian time series with a desired ("target") power spectrum and bispectrum is presented. Specifically, by generating a Gaussian time series (using amplitude information from the target power spectrum and random phases) and passing it through a quadratic filter (that uses phase information from the target bispectrum), a realization of a quadratically nonlinear random process with a specified power spectrum and bispectrum can be produced. Second- and third-order statistics from many realizations of simulated nonlinear time series compare well to those from the original time series providing the target power spectrum and bispectrum, with deviations consistent with theory. The simulation technique is shown to simulate accurately ocean waves in shallow water, which are well known to be quadratically nonlinear.
A Secure Communications System Using Chaotic Switching
Authors: Lee Chungyong , Williams Douglas B. , Lee Jaejin
Published: September 2003

Since previous secure communications methods using chaotic systems require very high SNRs, knowledge of the initial conditions, and/or chaotic systems with the self-synchronization property, their application is limited. In this paper, a new secure communications scheme based on chaotic switching is discussed. The proposed scheme makes decisions on the incoming bits by comparing the power level of the dynamical error for each symbol signal after applying a noise reduction method based on the known system dynamics. A multi-user secure communications scheme is also considered where each user's symbol signal is mapped through a unique scramble matrix. We also examine whether these scramble matrices help to combat against smart jamming. The proposed system does not require self-synchronization between the chaotic receiver and transmitter and exhibits reasonable performance at relatively low SNRs.
Higher-Order Spectral Analysis to Detect Power-Frequency Mechanisms in a Driven Chua's Circuit
Authors: Leenaerts Domine M. W.
Published: September 2003

Higher-order spectra have been used to investigate nonlinear interactions between frequency modes in a driven Chua's circuit. The spectra show that an energy transfer takes place to the dominant frequencies in the circuit, i.e. the input frequency, the primary peak frequency and the harmonics of both frequencies. Other frequencies couplings become less important. Obviously, powers are (nonlinearly) related at different frequencies. When the circuit undergoes a period doubling sequence to chaos, the gain is increasing.
Nonlinear Galerkin Methods for 3D Magnetohydrodynamic Equations
Authors: Schmidtmann Olaf , Feudel Fred , Seehafer Norbert
Published: September 2003

The usage of nonlinear Galerkin methods for the numerical solution of partial differential equations is demonstrated by treating an example. We describe the implementation of a nonlinear Galerkin method based on an approximate inertial manifold for the 3D magnetohydrodynamic equations and compare its efficiency with the linear Galerkin approximation. Special bifurcation points, time-averaged values of energy and enstrophy as well as Kaplan–Yorke dimensions are calculated for both schemes in order to estimate the number of modes necessary to correctly describe the behavior of the exact solutions.
Embedded Surfaces with Ergodic Geodesic Flows
Authors: Burns Keith , Donnay Victor J.
Published: September 2003

Following ideas of Osserman, Ballmann and Katok, we construct smooth surfaces with ergodic, and indeed Bernoulli, geodesic flow that are isometrically embedded in R3. These surfaces can have arbitrary genus and can be made analytic.
On the Mechanism of Spatial Bifurcations in the Open Flow System
Authors: Yamaguchi Akihiro
Published: September 2003

The phenomenon of spatial bifurcations in the open flow system (OFS), which is a unidirectionally coupled map lattice, is investigated. The bifurcation conditions are obtained by analyzing the effect of deterministic discretization of the system's state when the OFS is simulated by the computer. The analytical results are examined by application to the OFS with logistic maps.
Amplitude Equations and Chemical Reaction–Diffusion Systems
Authors: Ipsen M. , Hynne F. , Sørensen P. G.
Published: September 2003

The paper discusses the use of amplitude equations to describe the spatio-temporal dynamics of a chemical reaction–diffusion system based on an Oregonator model of the Belousov–Zhabotinsky reaction. Sufficiently close to a supercritical Hopf bifurcation the reaction–diffusion equation can be approximated by a complex Ginzburg–Landau equation with parameters determined by the original equation at the point of operation considered. We illustrate the validity of this reduction by comparing numerical spiral wave solutions to the Oregonator reaction–diffusion equation with the corresponding solutions to the complex Ginzburg–Landau equation at finite distances from the bifurcation point. We also compare the solutions at a bifurcation point where the systems develop spatio-temporal chaos. We show that the complex Ginzburg–Landau equation represents the dynamical behavior of the reaction–diffusion equation remarkably well, sufficiently far from the bifurcation point for experimental applications to be feasible.
Basin Fractalization Due to Focal Points in a Class of Triangular Maps
Authors: Bischi Gian-Italo , Gardini Laura
Published: September 2003

For a class of rational triangular maps of a plane, characterized by the presence of points in which a component assumes the form [Formula: see text], a new type of bifurcation is evidenced which creates loops in the boundaries of the basins of attraction. In order to explain such bifurcation mechanism, new concepts of focal point and line of focal values are defined, and their effects on the geometric behavior of the map and of its inverses are studied in detail. We prove that the creation of loops, which generally constitute the boundaries of lobes of the basins issuing from the focal points, is determined by contacts between basin boundaries and the line of focal values. A particular map is proposed for which the sequence of such contact bifurcations occurs, causing a fractalization of basin boundaries. Through the analytical and the numerical study of this example new structures of the basins of attraction are evidenced, characterized by fans of stable sets issuing from the focal points, assuming the shape of lobes and arcs, the latter created by the merging of lobes due to contacts between the basin boundaries and the critical curve LC.
Signal Extraction from Chaotic Communications
Authors: Short Kevin M.
Published: September 2003

This paper will consider the use of nonlinear dynamic (NLD) forecasting to extract messages from chaotic communication systems. Earlier work has shown that one-step prediction methods have sometimes been able to reveal the presence of hidden messages as well as the frequency content of the hidden messages. However, recovery of the actual hidden message usually involved filtering in the frequency domain. In this paper we show that it may be possible to extract the hidden message signal without filtering in the frequency domain. The approaches which will be discussed involve either the use of multi step predictions or a resumming process on the residuals after one-step prediction. For the multi step methods, two related approaches will be used. The first is primarily applicable to periodic signals, and will use the frequency information from one-step predictions to determine a block size to use for multi step predictions. The second will attempt dynamic detection of message signals using a windowed measure of the prediction error. For the resumming approach, it will be shown that even a lowest-order approximation can lead to faithful extraction of the hidden message signal for a ramping message signal that proved difficult in earlier work. These approaches will be applied to simple examples of communication schemes using signal masking and modulated chaos.
Misiurewicz Points for Complex Exponentials
Authors: Devaney Robert L. , Jarque Xavier
Published: September 2003

In this paper we examine the structure of the chaotic regime or Julia set of certain complex exponential maps Eλ(z) = λez. In the case where λ is a Misiurewicz point (i.e. the singular value 0 is eventually periodic), it is known that the Julia set for the map is the entire plane. In this case the Julia set also possesses certain curves or "hairs" that are permuted by the map. We examine the dynamics on these hairs in detail. We describe a certain extended symbolic dynamics by which the topological structure of the hairs may be determined completely.
Noninvertible Piecewise Linear Maps Applied to Chaos Synchronization and Secure Communications
Authors: Millerioux G. , Mira C.
Published: September 2003

Recently, it was demonstrated that two chaotic dynamical systems can synchronize each other, leading to interesting applications as secure communications. We propose in this paper a special class of dynamical systems, noninvertible discrete piecewise linear, emphasizing on interesting advantages they present compared with continuous and differentiable nonlinear ones. The generic aspect of such systems, the simplicity of numerical implementation, and the robustness to mismatch of technological parameters make them good candidates. The classical concept of controllability in the control theory is presented and used in order to choose and predict the number of appropriate variables to be transmitted for synchronization. A necessary and sufficient condition of chaotic synchronization is established without computing numerical quantities, introducing a state affinity structure of chaotic systems which provides an a priori establishment of synchronization.
Chaotic Synchronization Conditions Based on Control Theory for Systems Described by Discrete Piecewise Linear Maps
Authors: Millerioux G.
Published: September 2003

Many approaches have been developed in order to design synchronized chaotic systems. Meanwhile, the choice and the number of appropriate variables transmitted in an unidirectional coupling for the synchronization of chaotic systems were not clearly established. In this paper, such problems are handled by introducing the classical concepts of controllability, observability and state affinity of the control theory. Two distinct structures of synchronization are introduced and unified by a duality property. A state space representation is adopted and a necessary condition of synchronization is established providing the choice and the number of variables transmitted. Then, a necessary and sufficient condition is established which permits one to obtain a "global synchronization".
Adaptive Control of the Uncertain Duffing Oscillator
Authors: Dong Xiaoning , Chen Guanrong , Chen Liang
Published: September 2003

An adaptive feedback controller is developed based on rigorous Lyapunov argument for an uncertain chaotic Duffing oscillator, in which the three key system parameters are essentially unknown. The proposed method can be easily extended to handle some other chaotic dynamical systems with mild modifications. Simulation results are included to verify the control theory and controller design method.
Investigation of the Bifurcation Process in the RL Diode Circuit by Employing Mathieu and Hill Equations
Authors: Sarafian G. , Kaplan B. Z.
Published: September 2003

The work deals with the dynamic processes of the RL diode circuit at the bifurcation points. It is shown, that just at the first bifurcation point (slightly beyond it) the circuit dynamics can be modeled by the Mathieu equation. Furthermore, the role of the voltage source there is not in imposing forcing function to the system equation. It has been discovered that the role of the source is that of a pumping source in the parametric sense. Hence, the resonator behaves at the bifurcation point as if it were a resonator, whose reactor is periodically time-varying. Further investigations of the dynamics at higher bifurcation points reveal that a similar mechanism is repeated and the processes there can be modeled by the Hill equation (which is a generalized version of the Mathieu equation). The theoretical investigation for the first bifurcation point is supported by experiments. Practical half-tone generators are shown to be closely related to the RL diode circuit.
Detection of Tertiary Hopf Bifurcations Arising from Mode Interactions
Authors: Amdjadi F. , Aston P. J.
Published: September 2003

In the unfolding of a mode interaction, in addition to the primary bifurcations, there are also secondary bifurcations which occur on the primary branches giving rise to mixed mode solutions. A further tertiary Hopf bifurcation arises in some cases from the mixed mode solutions. The detection of Hopf bifurcation points is a numerically expensive procedure and so we consider whether it is possible to predict the existence of the tertiary Hopf bifurcation by considering only the geometric structure of the primary and secondary branches. We show that in some cases, it is possible to show that no Hopf bifurcation exists while in other cases, more information in the form of the stability of the trivial solution is required to determine whether or not the Hopf bifurcation exists. An algorithm for determining the existence of the Hopf bifurcation is given.
Links and Bifurcations in Nonsingular Morse–Smale Systems
Authors: Campos B. , Alfaro J. Martínez , Vindel P.
Published: September 2003

Wada's theorem classifies the set of periodic orbits in NMS systems on S3 as links, that can be written in terms of six operations.

This characterization allows us to study the topological restrictions that links require to suffer a given codimension one bifurcation.

Moreover, these results are reproduced in the case of NMS systems with rotational symmetries, introducing new geometrical tools.
Bifurcations and Synchronization using an Integrated Programmable Chaotic Circuit
Authors: Delgado-Restituto M. , Liñán M. , Ceballos J. , Rodríguez-Vázquez A.
Published: September 2003

This paper presents a CMOS chip which can act as an autonomous stand-alone unit to generate different real-time chaotic behaviors by changing a few external bias currents. In particular, by changing one of these bias currents, the chip provides different examples of a period-doubling route to chaos. We present experimental orbits and attractors, time waveforms and power spectra measured from the chip. By using two chip units, experiments on synchronization can be carried out as well in real-time. Measurements are presented for the following synchronization schemes: linear coupling, drive-response and inverse system. Experimental statistical characterizations associated to these schemes are also presented. We also outline the possible use of the chip for chaotic encryption of audio signals. Finally, for completeness, the paper includes also a brief description of the chip design procedure and its internal circuitry.
Pulses, Fronts and Chaotic Wave Trains in a One-Dimensional Chua's Lattice
Authors: Kazantsev V. B. , Nekorkin V. I. , Velarde M. G.
Published: September 2003

We show how wave motions propagate in a nonequilibrium discrete medium modeled by a one-dimensional array of diffusively coupled Chua's circuits. The problem of the existence of the stationary wave solutions is reduced to the analysis of bounded trajectories of a fourth-order system of nonlinear ODEs. Then, we study the homoclinic and heteroclinic bifurcations of the ODEs system. The lattice can sustain the propagation of solitary pulses, wave fronts and complex wave trains with periodic or chaotic profile.
Bifurcations and Chaos in a Linear Control System with Saturated Input
Authors: Alvarez Joaquín , Curiel Luis E.
Published: September 2003

An analysis of the dynamical behavior of a saturated input, second-order, linear plant feed back with a simple proportional controller is presented. The synthesized control signal is passed through a saturator before being applied to the plant. Existence and stability of equilibria depending on the controller gain are analyzed. Existence of periodic and homoclinic orbits is also discussed. It is shown that a periodic disturbance and a negative gain can cause the closed-loop system to have a strange (chaotic) invariant set. Numerical and experimental results illustrate the analysis presented.
Modeling and Predicting Non-Stationary Time Series
Authors: Cao Liangyue , Mees Alistair , Judd Kevin
Published: September 2003

Many experimental time series are non-stationary. Modeling and predicting them is generally considered to be difficult. In this paper we introduce time-dependent regressive (TDR) models, which depend not only on system states but also on time. We test artificial time series which come from parameter-changing systems and are therefore non-stationary, and a simulated experimental time series from a model of a non-stationary industrial system. The TDR models work well on those time series, not only in prediction but also in extraction of the underlying bifurcations.
Turbulent Behavior of Stock Exchange Indices and Foreign Currency Exchange Rates
Authors: Hilgers Angela , Beck Christian
Published: September 2003

The time evolution of stock exchange indices and foreign currency exchange rates has many similarities with turbulent flows. In particular, the probability densities of price changes are non-Gaussian and develop stretched exponential tails, quite similar to the densities of velocity differences measured in fully developed hydrodynamical turbulence. We show that a simple cascade model, based on a self-similar, hierarchical dynamics of price changes, describes the observed probability densities of the financial indices in a quantitatively correct way.
On the Definitions of Bifurcation
Authors: Zhou K. , Wu J. K.
Published: September 2003

In this paper the equivalence of two definitions of elementary bifurcation is proved and a new definition for bifurcation is given.
Period-Adding Bifurcation with Chaos in the Interspike Intervals Generated by an Experimental Neural Pacemaker
Authors: Ren W. , Hu S. J. , Zhang B. J. , Wang F. Z. , Gong Y. F. , Xu J. X.
Published: September 2003

The dynamics of the generation of the various spike trains in neural pacemakers is of fundamental importance to the understanding of neural coding. Recent studies have demonstrated, theoretically and experimentally, that neural pacemakers produce chaotic oscillations. Deeper analyses in several neuronal models have revealed many nonlinear phenomena including period-adding bifurcations whose existence has not been experimentally confirmed. In this letter, we reported that the period-adding bifurcation with chaos was observed in the interspike interval (ISI) series generated by an experimental neural pacemaker when the extracellular calcium concentration was changed or a potassium channel blocker was administered at the site of the pacemaker. We also simulated our experimental discoveries by computing a generalized model of excitable cells. The chaotic phenomenon in the experiment and that in the model were demonstrated and compared using the nonlinear forecasting and surrogate data methods.
Self-Organization Processes in Chaotic Neural Networks Under External Periodic Force
Authors: Bondarenko Vladimir E.
Published: September 2003

Self-organization processes in an analog asymmetric neural network with time delay and under an external sinusoidal force are considered. Quantitative characteristics of the neuron outputs (spectrum, correlation dimension, largest Lyapunov exponent, Shannon entropy, normalized and renormalized Shannon entropies) are studied in dependence on the frequency and amplitude of the external force. It is shown that the external sinusoidal force allows the control of the degree of chaos and produces transitions "order–chaos", "chaos–order" and "chaos–chaos" with different quantitative characteristics. Information processing both by the individual neurons and by the neural network as a system is discussed. Chaotic neural network under an external force is considered as a qualitative model of the infra-frequencies action on the brain.
Bifurcation and Chaotic Behaviors in a Discrete Variable Structure System with Unbounded Control Magnitude
Authors: Yu Xinghuo
Published: September 2003

Bifurcation and chaotic behaviors in a discrete variable structure system with unbounded control magnitude are examined in this paper. The analysis using the exponential discretization shows that the sampling period is actually an additional degree of freedom whose value is crucial to the occurrence of bifurcation and chaotic behaviors. The bifurcation point of the sampling period is derived. Moreover, it is shown that other commonly used approximate discretization schemes such as zero-order-hold may cause similar problems as well. The bifurcation points of sampling periods for these schemes are an approximation of that for the exponential discretization scheme. Computer simulations are presented to illustrate the bifurcation and chaotic phenomena.
Chua's Circuit and the Qualitative Theory of Dynamical Systems
Authors: Mira Christian
Published: September 2003

Simple electronic oscillators were at the origin of many studies related to the qualitative theory of dynamical systems. Chua's circuit is now playing an equivalent role for the generation and understanding of complex dynamics.

In honour of my friend Leon Chua on his 60th birthday.
Mathematical Problems of Nonlinear Dynamics: A Tutorial
Authors: Shilnikov Leonid
Published: September 2003

We review the theory of nonlinear systems, especially that of strange attractors, and give its perspectives. Of a special attention are the recent results concerning hyperbolic attractors and features of high-dimensional systems in the Newhouse regions. We present an example of a "wild" strange attractor of the topological dimension three.
Experimental Nonlinear Physics
Authors: Lauterborn Werner , Kurz Thomas , Parlitz Ulrich
Published: September 2003

The review gives and account of the historical development, the current state and possible future developments of experimental nonlinear physics, with emphasis on acoustics, hydrodynamics and optics. The concepts of nonlinear time-series analysis which are the basis of the analysis of experimental outcomes from nonlinear systems are explained and recent developments pertaining to such different fields as modeling, prediction, nonlinear noise reduction, detecting determinism, synchronization, and spatio-temporal time series are surveyed. An overview is given of experiments on acoustic cavitation, a field rich of nonlinear phenomena such as nonlinear oscillations, chaotic dynamics and structure formation, and one of the first physical systems to exhibit period-doubling and chaos in experiment.
Nonlinear Physics: Integrability, Chaos and Beyond
Authors: Lakshmanan M.
Published: September 2003

Integrability and chaos are two of the main concepts associated with nonlinear physical systems which have revolutionized our understanding of them. Highly stable exponentially localized solitons are often associated with many of the important integrable nonlinear systems while motions which are sensitively dependent on initial conditions are associated with chaotic systems. Besides dramatically raising our perception of many natural phenomena, these concepts are opening up new vistas of applications and unfolding technologies: Optical soliton based information technology, magnetoelectronics, controlling and synchronization of chaos and secure communications, to name a few. These developments have raised further new interesting questions and potentialities. We present a particular view of some of the challenging problems and payoffs ahead in the next few decades by tracing the early historical events, summarizing the revolutionary era of 1950–1970 when many important new ideas including solitons and chaos were realized and reviewing the current status. Important open problems both at the basic and applied levels are discussed.
Nonlinear Computation
Authors: Seydel Rüdiger
Published: September 2003

Nonlinear Computation will provide a framework for the computational science and engineering of the future. Topics such as bifurcation and chaos, and methods such as continuation and branch switching, and features such as stability and sensitivity are basic ingredients in the attempt to understand our world. Nonlinear Computation is indispensable when we try to predict and simulate the dynamics of states in the biological systems we consist of, in the ecological and economical systems we live in, and in the technical systems we make use of.

This paper essentially consists of three parts. The first part (Sec. 1) explains what nonlinear computation means. The second part (Secs. 2–5) presents a survey on nonlinear phenomena, and on the methods of nonlinear computation. The final part (Secs. 6 and 7) will speculate on visions and trends for the next decades.
Nonlinear Numerics
Authors: Doedel Eusebius
Published: September 2003

The objectives and some basic methods of numerical bifurcation analysis are described. Several computational examples are used to illustrate the power as well as the limitations of these techniques. Directions in which further algorithmic and software development is desirable are also discussed.
Some Historical Aspects of Nonlinear Dynamics: Possible Trends for the Future
Authors: Mira Christian
Published: September 2003

This paper does not pretend to present a comprehensive history of nonlinear dynamics. Its purpose is more modest and limited to some historical aspects of this topic. The first part of this paper deals with the early foundations of nonlinear dynamics (essentially the Poincaré and Lyapunov results). The succeeding sections cover the period 1910–1970 and describes the development and contributions of the theory, elaborated by Birkhoff, the Andronov school, and the Krylov–Bogoliubov school. After 1970, the development of new results in nonlinear dynamics has become "explosive". A part of these results is presented in a summarized form in this paper. The last section suggests some possible trends for future research.
Control and Applications of Chaos
Authors: Grebogi Celso , Lai Ying–Cheng , Hayes Scott
Published: September 2003

This review describes a procedure for stabilizing a desirable chaotic orbit embedded in a chaotic attractor of dissipative dynamical systems by using small feedback control. The key observation is that certain chaotic orbits may correspond to a desirable system performance. By carefully selecting such an orbit, and then applying small feedback control to stabilize a trajectory from a random initial condition around the target chaotic orbit, desirable system performance can be achieved. As applications, three examples are considered: (1) synchronization of chaotic systems; (2) conversion of transient chaos into sustained chaos; and (3) controlling symbolic dynamics for communication. The first and third problems are potentially relevant to communication in engineering, and the solution of the second problem can be applied to electrical power systems to avoid catastrophic events such as the voltage collapse.
Chaos: Generating Complexity from Simplicity
Authors: Brown Ray , Chua Leon O.
Published: September 2003

The most commonly used mapping to illustrate the phenomenon of chaos is the map x → 2x mod (1). This map is known as the 'unilateral shift' because, in the binary number system this map shifts all digits to the left by one decimal place, and truncates the integer. The second most commonly used paradigm of chaos is the Smale horseshoe whose complexity is essentially the bilateral shift obtained when we simply shift without truncation in some symbol system. Neither of these paradigms fully explains chaos since shifts cannot generate complex orbits from simple (rational) initial conditions. How chaos generates complexity from simplicity is an essential part that needs explanation. Providing this explanation is the objective of this paper.
A Turning Point Analysis of the Ergodic Dynamics of Iterative Maps
Authors: Schmelcher P. , Diakonos F. K.
Published: September 2003

The dynamics of one-dimensional iterative maps in the regime of fully developed chaos is studied in detail. Motivated by the observation of dynamical structures around the unstable fixed point we introduce the geometrical concept of a turning point which represents a local minimum or maximum of the trajectory. Following we investigate the highly organized and structured distribution of turning points. The turning point dynamics is discussed and the corresponding turning point map which possesses an appealing asymptotic scaling property is investigated. Strong correlations are shown to exist for the turning point trajectories which contain the information of the fixed points as well as the stability coefficients of the dynamical system. For the more specialized case of symmetric maps which possess a symmetric density we derive universal statistical properties of the corresponding turning point dynamics. Using the turning point concept we finally develop a method for the analysis of (one-dimensional) time series.
Almost Invariant Sets in Chua's Circuit
Authors: Dellnitz Michael , Junge Oliver
Published: September 2003

Recently multilevel subdivision techniques have been introduced in the numerical investigation of complicated dynamical behavior. We illustrate the applicability and efficiency of these methods by a detailed numerical study of Chua's circuit. In particular we will show that there exist two regions in phase space which are almost invariant in the sense that typical trajectories stay inside each of these sets on average for quite a long time.
Using Generalized Cell Mapping to Approximate Invariant Measures on Compact Manifolds
Authors: Guder Rabbijah , Kreuzer Edwin
Published: September 2003

In order to predict the long term behavior of nonlinear dynamical systems the generalized cell mapping is an efficient and powerful method for numerical analysis. For this reason it is of interest to know under what circumstances dynamical quantities of the generalized cell mapping (like persistent groups, stationary densities, …) reflect the dynamics of the system (attractors, invariant measures, …). In this article we develop such connections between the generalized cell mapping theory and the theory of nonlinear dynamical systems. We prove that the generalized cell mapping is a discretization of the Frobenius–Perron operator. By applying the results obtained for the Frobenius–Perron operator to the generalized cell mapping we outline for some classes of transformations that the stationary densities of the generalized cell mapping converges to an invariant measure of the system. Furthermore, we discuss what kind of measures and attractors can be approximated by this method.
First Harmonic Analysis of Linear Control Systems with High-Gain Saturating Feedback
Authors: Aguirre Baltazar , Alvarez-Ramírez José , Fernández Guillermo , Suárez Rodolfo
Published: September 2003

A first harmonic approach (describing function method) is used in this work to study the behavior of linear control systems with a saturating high-gain linear feedback. It is shown that when the eigenvalues of the closed-loop system are located deeper in the left-half complex plane, several unstable periodic orbits shrink to the origin, thus leading to an unstable equilibrium point. This dynamical behavior is interpreted as the vanishing of the region of attraction of the origin when a saturating high-gain feedback is used.
Experiments on RF Band Communications Using Chaos
Authors: Dmitriev A. S. , Panas A. I. , Starkov S. O. , Kuzmin L. V.
Published: September 2003

This paper presents experimental results of wire-line and wireless analog information transmission in RF band via chaos using standard signal formation and reception methods.
Devaney Chaos in an Approximate One-Dimensional Model of the Colpitts Oscillator
Authors: O'Cairbre Fiacre , Maggio Gian Mario , Kennedy Michael Peter
Published: September 2003

In this paper we consider an approximate one-dimensional model for describing the non-linear dynamics of the Colpitts oscillator. It has already been shown that this model preserves the qualitative dynamical behavior of the Colpitts oscillator. Using the theory of snap-back repellors we prove that the one-dimensional model is chaotic for certain values of its parameters.
Chaos in the Portevin–Le Châtelier Effect
Authors: Noronha S. J. , Ananthakrishna G. , Quaouire L. , Fressengeas C. , Kubin L. P.
Published: September 2003

We report the verification of the prediction of chaos in the Portevin–Le Châtelier effect or the jerky flow by analyzing the stress signals obtained from samples of polycrystalline Al–Mg alloys subjected to a constant strain rate test. Particular care is taken to obtain reasonably long and accurate stress signals. The analysis of these signals is carried out by using several complementary methods such as calculation of correlation dimension, singular value decomposition and the spectrum of Lyapunov exponents. The analysis shows the existence of a finite correlation dimension and a positive Lyapunov exponent. Using the existence of a positive Lyapunov exponent and finite correlation dimension as a discriminator, we also carry out a surrogate analysis of the time series to ascertain that the signals are not from a power law stochastic process. The analysis provides an unambiguous support for the existence of chaos in Portevin–Le Châtelier effect thus verifying the prediction of the model. Further, from the analysis we find that the minimum number of variables required for a dynamical description of the jerky flow appears to be four or five consistent with the model.
Obtaining Synchronization for a Class of Chaotic Systems
Authors: Sharma Naresh , Poonacha P. G.
Published: September 2003

An iterative algorithm is proposed for constructing synchronizing receivers for a chosen class of chaotic transmitter systems. A complete knowledge of the transmitter system is not required. Identical synchronization is ensured with the transmitted signal in the constructed receivers. Potential applications of the method are discussed for communications.
On Identifying Global Nonlinear Discrete Models from Chaotic Data
Authors: Mendes Eduardo M. A. M. , Billings S. A.
Published: September 2003

This paper investigates the identification of global discrete nonlinear models from chaotic data. It is shown that when chaotic data from a nonlinear system do not contain enough information about all the fixed points, the usual model selection procedures can select different local nonlinear models.
Circle Maps and the Devil's Staircase in a Periodically Perturbed Oregonator
Authors: Brøns Morten , Gross Peter , Bar-Eli Kedma
Published: September 2003

Markman and Bar-Eli has studied a periodically forced Oregonator numerically and found a parameter range with the following properties: (1) Only periodic solutions are found in frequency-locked steps, each with a certain pattern of large and small oscillations (2) Between any two steps there is a step with the period being the sum of the two periods and the concatenation of the two patterns (3) Certain scaling properties as the period tends to infinity. We show that such behavior occurs if the dynamics of the system is governed by a family of diffeomorphisms of a circle with a Devil's staircase. Using invariant manifold theory we argue that an invariant circle must indeed exist when, as in the present case, the unforced system is close to a saddle-loop bifurcation. Generalizations of the results are briefly discussed.
Dynamical Analysis of Time Series by Statistical Tests
Authors: Deco Gustavo , Schittenkopf Christian , Schürmann Bernd
Published: September 2003

In this review we deal with the application of statistical test techniques for the extraction of structures in time series. Two kinds of questions are answered in this statistical framework: Are there any temporal dependences in the data? and Which kind of dynamics generate these temporal dependences? The first question is known as the problem of predictability and also considers the aspect of stationarity. The second question is deeper in the sense that it deals with the dynamical characterization of the detected temporal structures. Central to our approach is a cumulant-based measure of statistical dependences in Fourier space. The dynamical aspects are studied by means of the information flow. The theory is illustrated by artificial and real-world, stochastic and chaotic examples.
Numerical Studies of the Periodically Forced Bonhoeffer van der Pol System
Authors: Barnes Belinda , Grimshaw Roger
Published: September 2003

The Bonhoeffer van der Pol system, with an applied constant forcing, was invoked by Fitzhugh [1961] as a two-dimensional representation of the four-dimensional Hodgkin–Huxley system [1952], a well-known physiological model representing the electrical behavior across a nerve membrane. The system has been analyzed within a particular parameter regime relevant to the physiology (see [Fitzhugh, 1961]) and for the full parameter space with emphasis on the prediction of periodic solutions (see [Barnes & Grimshaw, 1995]). In this paper the system is considered with a time dependent sinusoidal forcing term in which form it represents a nonlinear, nonautonomous system of differential equations with five parameters. The study is motivated by physiological experiments with neurons subjected to periodic stimuli (see, e.g. [Hayashi et al., 1982]). A few aspects pertaining to the system behavior have been explored by others for particular fixed parameter combinations (and with different purposes from those here), for example [Wang, 1989; Rajasekar & Lakshmanan, 1988, 1993; Yasin et al., 1993; Braaksma & Grasman, 1993; Rabinovitch et al., 1994]). In this paper we present the numerical results of simulations for a more general parameter space and propose theoretical interpretations for a broad range of the behavior states incurred. Further, we present numerical results describing how the system dynamics change as each of the five parameters is varied, and thus predict regions where periodic, quasiperiodic or chaotic behavior can be expected.
The Role of Period Tripling in the Development of a Self Similar Bifurcation Structure
Authors: Goswami Binoy Krishna
Published: September 2003

We study theoretically the period tripling phenomenon and its role in the development of the bifurcation structure of the periodically forced Toda oscillator. The analytical studies reveal some interesting features in the context of previously known period tripling in one-parameter systems where the loss of stability of the original equilibrium used to be associated with the birth of a bifurcating saddle. In the case of Toda oscillator, a period-tripled saddlenode is born around the stable original period where the saddle approaches the original period as the dissipativity is reduced. In the conservative limit the saddle undergoes a tangential interaction with the original centre. The recurrent appearance of period tripling and doubling suggest a new self similar feature of the associated bifurcation structure. The revelation of the underlying period tripling phenomena also helps to sequentially characterize each period, cascade and substructure, and to make some qualitative predictions regarding their locations in the phase and parameter space. The period tripling may give birth to some "degenerate period n orbits" (i.e. a certain class of period n orbits with identical n). However, they are distinctly characterized on the basis of concept of period tripling and we also predict the "degeneracy" (the number of such period n orbits for a given n) analytically. The same concept is extended in the case of degenerate cascades and substructures.
Structure and Breakdown of Invariant Tori in a 4-D Mapping Model of Accelerator Dynamics
Authors: Vrahatis M. N. , Isliker H. , Bountis T. C.
Published: September 2003

We study sequences of periodic orbits and the associated phase space dynamics in a 4-D symplectic map of interest to the problem of beam stability in circular particle accelerators. The increasing period of these orbits is taken from a sequence of rational approximants to an incommensurate pair of irrational rotation numbers of an invariant torus. We find stable (elliptic–elliptic) periodic orbits of very high period and show that smooth rotational tori exist in their neighborhood, on which the motion is regular and bounded at large distances away from the origin. Perturbing these tori in parameter and/or initial condition space, we find either chains of smaller rotational tori or certain twisted tube-like tori of remarkable morphology. These tube-tori and tori chains have small scale chaotic motions in their surrounding vicinity and are formed about invariant curves of the 4-D map, which are either single loops or are composed of several disconnected loops, respectively. These smaller chaotic regions as well as the non-smoothness properties of large rotational tori under small perturbations, leading to eventual escape of orbits to infinity, are studied here by the computation of correlation dimension and Lyapunov exponents.
The Dynamics of Symmetric Bimodal Maps
Authors: Lofaro Thomas
Published: September 2003

The dynamics and bifurcations of a family of odd, symmetric, bimodal maps, fαare discussed. We show that for a large class of parameter values the dynamics of fαcan be described via an identification with a unimodal map uα. In this parameter regime, a periodic orbit of period 2n + 1 of uαcorresponds to a periodic orbit of period 4n + 2 for fα. A periodic orbit of period 2n of uαcorresponds to a pair of distinct periodic orbits also of period 2n for fα. In a more general setting we describe the genealogy of periodic orbits in the family fαusing symbolic dynamics and kneading theory. We identify which periodic orbits of even periods are born in period-doubling bifurcations and which are born in pitchfork bifurcations and provide a method of describing the "ancestors" and "descendants" of these orbits. We also show that certain periodic orbits of odd periods are born in saddle-node bifurcations.
Computation of Periodic Solutions of Differential-Algebraic Equations in the Neighborhood of Hopf Bifurcation Points
Authors: Zheng Q. , Neubert R.
Published: September 2003

A new method for the computation of periodic solutions of differential-algebraic equations (DAEs) in the neighborhood of Hopf bifurcation points of DAEs is proposed. The method is based on an asymptotic energy analysis of the dynamic behavior near the Hopf bifurcation point. The algorithm is described in more detail in combination with the method of harmonic balance which is used to compute the periodic solutions. Furthermore the usability of the method also in the case of nonclassical Hopf bifurcation points is explained. Finally numerical results using an implementation of the method in an electrical circuit simulator are given.
Chaotic Digital Code-Division Multiple Access (CDMA) Communication Systems
Authors: Yang Tao , Chua Leon O.
Published: September 2003

In this paper, the structure, principle and framework of chaotic digital code-division multiple access ((CD)2 MA) communication systems are presented. Unlike the existing CDMA systems, (CD)2MA systems use continuous pseudo-random time series to spread the spectrum of message signal and the spread signal is then directly sent through a channel to the receiver. In this sense, the carrier used in (CD)2MA is a continuous pseudo-random signal instead of a single tone as used in CDMA. We give the statistical properties of the noise-like carriers. In a (CD)2MA system, every mobile station has the same structure and parameters, only different initial conditions are assigned to different mobile stations. Instead of synchronizing two binary pseudo-random sequences as in CDMA systems, we use an impulsive control scheme to synchronize two chaotic systems in (CD)2MA. The simulation results show that the channel capacity of (CD)2MA is twice as large than that of CDMA.
Controlling Chaos in a Pendulum Subjected to Feedforward and Feedback Control
Authors: Yagasaki Kazuyuki , Uozumi Tomotsugu
Published: September 2003

We consider a pendulum subjected to feedforward and feedback control, in which chaotic motions occur when the feedback gain is small. We apply two control techniques recently proposed by Ott and his coworkers so that the pendulum can exhibit the desired motion. The techniques use some dynamical properties relating to chaos. Numerical examples are given and the effectiveness of these control techniques is demonstrated.