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Showing papers in "International Journal of Bifurcation and Chaos in 1999"


Journal ArticleDOI
TL;DR: In this paper, the authors reported the finding of a chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.
Abstract: This Letter reports the finding of a new chaotic at tractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler at tractors.

2,443 citations


Journal ArticleDOI
TL;DR: In this paper, the authors show that many coupled oscillator array configurations can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function which solves, once and for all, the problem of synchronous stability for many couplings of that oscillator.
Abstract: We show that many coupled oscillator array configurations considered in the literature can be put into a simple form so that determining the stability of the synchronous state can be done by a master stability function which solves, once and for all, the problem of synchronous stability for many couplings of that oscillator.

411 citations


Journal ArticleDOI
TL;DR: A theorem presents the necessary and sufficient conditions under which a linear GS can be achieved between two chaotic systems and studies the linear GS of two Chua's circuits.
Abstract: Generalized synchronization (GS) of two chaotic systems is a generalization of identical synchronization. Usually, the manifold of GS is much more complex than the driven system and the driving system. In this paper, we study a special case of GS in which the synchronization manifold is linear (linear GS for short). In a theorem, we present the necessary and sufficient conditions under which a linear GS can be achieved between two chaotic systems. In particular, we study the linear GS of two Chua's circuits.

159 citations


Journal ArticleDOI
TL;DR: In this paper, it is shown that once a proper set of feedback circuits is present in the Jacobian matrix of the system, the chaotic character of trajectories is remarkably robust versus changes in the nature of the nonlinearities.
Abstract: This paper aims to show how complex nonlinear dynamic systems can be classified, analyzed and synthesized in terms of feedback circuits. The Rossler equations for deterministic chaos are revisited and generalized in this perspective. It is shown that once a proper set of feedback circuits is present in the Jacobian matrix of the system, the chaotic character of trajectories is remarkably robust versus changes in the nature of the nonlinearities. "Labyrinth chaos", whereby simple differential systems generate large lattices of many unstable steady states embedded in a chaotic attractor, is constructed using this technique. In the limit case of a single three-element circuit without diagonal elements, one finds systems possessing an infinite lattice of unstable steady states between which trajectories percolate in a deterministic chaotic way.

130 citations


Journal ArticleDOI
TL;DR: In this paper, the authors discuss experimental demonstrations of chaotic communication in several optical systems and demonstrate chaotic communications through 35 km of single-mode optical fiber at up to 250 Mbit/s, a rate that is, at present, limited only by the speed of their detector electronics.
Abstract: We discuss experimental demonstrations of chaotic communication in several optical systems. In each, an erbium-doped fiber ring laser (EDFRL) produces chaotic fluctuations of light intensity onto which is modulated a message consisting of a sequence of pseudorandom digital bits. This combination of chaos and message propagates at a wavelength of ~ 1.5 microns through standard single-mode optical fiber from the transmitter to a receiver, where the message is recovered from the chaos. We present evidence of the high-dimensional nature of the chaotic waveforms and demonstrate chaotic communications through 35 km of single-mode optical fiber at up to 250 Mbit/s, a rate that is, at present, limited only by the speed of our detector electronics.

118 citations


Journal ArticleDOI
TL;DR: In this paper, it was shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillat oscillations. But the problem of destabilization of such systems was not addressed.
Abstract: The paper deals with the problem of destabilization of diffusively coupled identical systems. It is shown that globally asymptotically stable systems being diffusively coupled, may exhibit oscillat ...

115 citations


Journal ArticleDOI
TL;DR: In this article, the authors explore the dynamics of fronts in elastic excitable media and show that low-dimensional structures including synchronous oscillations and propagating fronts are dominant, in agreement with the results of laboratory friction experiments.
Abstract: The Burridge–Knopoff model of earthquake faults with viscous friction is equivalent to a van der Pol–FitzHugh–Nagumo model for excitable media with elastic coupling. The lubricated creep–slip friction law we use in Burridge–Knopoff model describes the frictional sliding dynamics of a range of real materials. Low-dimensional structures including synchronous oscillations and propagating fronts are dominant, in agreement with the results of laboratory friction experiments. Here we explore the dynamics of fronts in elastic excitable media.

111 citations


Journal ArticleDOI
TL;DR: This paper reviews the recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions.
Abstract: A cellular neural/nonlinear network (CNN) [Chua, 1998] is a biologically inspired system where computation emerges from a collection of simple nonlinear locally coupled cells. This paper reviews our recent research results beginning from the standard uncoupled CNN cell which can realize only linearly separable local Boolean functions, to a generalized universal CNN cell capable of realizing arbitrary Boolean functions. The key element in this evolutionary process is the replacement of the linear discriminant (offset) function w(σ)=σ in the "standard" CNN cell in [Chua, 1998] by a piecewise-linear function defined in terms of only absolute value functions. As in the case of the standard CNN cells, the excitation σ evaluates the correlation between a given input vector u formed by the outputs of the neighboring cells, and a template vector b, which is interpreted in this paper as an orientation vector. Using the theory of canonical piecewise-linear functions [Chua & Kang, 1977], the discriminant function is found to guarantee universality and its parameters can be easily determined. In this case, the number of additional parameters and absolute value functions m is bounded by m<2n-1, where n is the number of all inputs (n=9 for a 3×3 template). An even more compact representation where m

105 citations


Journal ArticleDOI
TL;DR: In this paper, a simplified neural network model for a ring of four neurons where each neuron receives two time delayed inputs: one from itself and another from the previous neuron is considered.
Abstract: We consider a simplified neural network model for a ring of four neurons where each neuron receives two time delayed inputs: One from itself and another from the previous neuron. Local stability analysis of the positive equilibrium leads to a characteristic equation containing products of four transcendental functions. By analyzing the equivalent system of four scalar transcendental equations, we obtain sufficient conditions for the linear stability of the positive equilibrium. Furthermore, we show that a Hopf bifurcation can occur when the positive equilibrium loses stability.

83 citations


Journal ArticleDOI
Makoto Itoh1
TL;DR: Some computer simulations and performance analysis are given to examine the validity of this scheme which transmits both analog and binary data via chaotic carriers through chaotic carriers.
Abstract: A new scheme is proposed for spread spectrum communication which transmits both analog and binary data via chaotic carriers. The proposed systems have some standard properties of spread spectrum communication. Some computer simulations and performance analysis are given to examine the validity of this scheme.

79 citations


Journal ArticleDOI
TL;DR: An observer-based approach to synthesis of synchronized chaotic systems using an observer for the linear system and a new concept of synchronization called (M, c)-synchronization is introduced.
Abstract: This paper proposes an observer-based approach to synthesis of synchronized chaotic systems. We consider a linear system with nonlinear feedback which exhibits chaos. Using an observer for the linear system, we design synchronized chaotic systems. Moreover, we introduce a new concept of synchronization called (M, c)-synchronization. We also propose a synthesis method for (M, c)-synchronized systems by modifying the observer-based method.

Journal ArticleDOI
TL;DR: In this article, the authors introduce the concept of focal points and prefocal curves to characterize some new kinds of contact bifurcations specific to maps with a vanishing denominator, which gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can only be observed if a map or some of its inverses has a vanishing numerator.
Abstract: This paper is devoted to the study of some global dynamical properties and bifurcations of two-dimensional maps related to the presence, in the map or in one of its inverses, of a vanishing denominator. The new concepts of focal points and of prefocal curves are introduced in order to characterize some new kinds of contact bifurcations specific to maps with denominator. The occurrence of such bifurcations gives rise to new dynamic phenomena, and new structures of basin boundaries and invariant sets, whose presence can only be observed if a map (or some of its inverses) has a vanishing denominator.

Journal ArticleDOI
TL;DR: The presented algorithms of encryption and decryption are based on multiple iteration of a certain dynamical chaotic system coming from gas dynamics models, utilizing a nonpredictability property of discrete chaotic systems.
Abstract: In the paper we propose a method of constructing cryptosystems, utilizing a nonpredictability property of discrete chaotic systems. We point out the requirements for such systems to ensure their security. The presented algorithms of encryption and decryption are based on multiple iteration of a certain dynamical chaotic system coming from gas dynamics models. A plaintext message specifies a part of the initial condition of the system (a particle's initial position). A secret key specifies the remaining part of initial condition (the particle's initial angle) as well as a sequence of discrete choices of the pre-images in the encryption procedure. We also discuss problems connected with the practical realization of such chaotic cryptosystems. Finally we demonstrate numerical experiments illustrating the basic properties of the proposed cryptosystem.

Journal ArticleDOI
TL;DR: In this article, backstepping design is proposed for controlling chaotic systems, which consists in a recursive procedure that combines the choice of a Lyapunov function with the design of feedback control.
Abstract: In this Letter backstepping design is proposed for controlling chaotic systems. The tool consists in a recursive procedure that combines the choice of a Lyapunov function with the design of feedback control. The advantages of the method are the following: (i) it represents a systematic procedure for controlling chaotic or hyperchaotic dynamics; (ii) it can be applied to several circuits and systems reported in literature; (iii) stabilization of chaotic motion to a steady state as well as tracking of any desired trajectory can be achieved. In order to illustrate the general applicability of backstepping design, the tool is utilized for controlling the chaotic dynamics of the Lorenz system and Chua's circuit. Finally, numerical simulations are carried out to show the effectiveness of the technique.

Journal ArticleDOI
TL;DR: In this paper, the authors show that the rapid bifurcation described by Kriegsmann [1987] is a generic bifurbation for planar symmetric piecewise-linear systems, which is responsible for the abrupt appearance of stable periodic oscillations.
Abstract: The rapid bifurcation described by Kriegsmann [1987] is shown to be a generic bifurcation for planar symmetric piecewise-linear systems. The bifurcation can be responsible for the abrupt appearance of stable periodic oscillations. Although it has some similarities with the Hopf bifurcation for smooth systems, since the stability change of an equilibrium involves the appearance of one limit cycle, the dependence of the limit cycle amplitude on the bifurcation parameter is different from the Hopf's case. To characterize this bifurcation, accurate estimates for the amplitude and period of the bifurcating limit cycle are given. The analysis is just illustrated with the application of the theoretical results to the Wien bridge oscillator. Comparisons with experimental data and Kriegsmann's analysis are also included.

Journal ArticleDOI
TL;DR: In this article, the authors predict stick-slip chaotic dynamics in a one-degree-of-freedom very weakly forced (quasiautonomous) oscillator using the Melnikov's technique.
Abstract: In this paper we predict stick–slip chaotic dynamics in a one-degree-of-freedom very weakly forced (quasiautonomous) oscillator using the Melnikov's technique. Numerical simulation confirms the validity of our approach.

Journal ArticleDOI
TL;DR: In this paper, it was shown that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under Eλ tend to infinity and hence are in the Julia set.
Abstract: In this paper we consider both the dynamical and parameter planes for the complex exponential family Eλ(z)=λez where the parameter λ is complex. We show that there are infinitely many curves or "hairs" in the dynamical plane that contain points whose orbits under Eλ tend to infinity and hence are in the Julia set. We also show that there are similar hairs in the λ-plane. In this case, the hairs contain λ-values for which the orbit of 0 tends to infinity under the corresponding exponential. In this case it is known that the Julia set of Eλ is the entire complex plane.

Journal ArticleDOI
TL;DR: The experimental results show that the accuracy of impulsive synchronization depends on both the period and the width of the impulse, and the ratio between the impulse width and impulse period for \almost-identical" synchronization increases as the impulse period increases.
Abstract: In this paper, experimental results on impulsive synchronization of two kinds of chaotic circuits; namely, Chua’s oscillator and a hyperchaotic circuit, are presented. To impulsively synchronize two Chua’s oscillators, synchronization impulses sampled from one state variable of the driving circuit are transmitted to the driven circuit. To impulsively synchronize two hyperchaotic circuits, synchronizing impulses sampled from two signals of the driving circuit are sent to the driven circuit. Our experimental results show that the accuracy of impulsive synchronization depends on both the period and the width of the impulse. The ratio between the impulse width and impulse period for \almost-identical" synchronization increases as the impulse period increases. The robustness of impulsive synchronization to additive noise is also experimentally studied. For suciently short impulse periods, no signicant dierences are observed between impulsive and continuous synchronizations. The performance of chaotic spread spectrum communication systems based on impulsive synchronization is also studied experimentally.

Journal ArticleDOI
TL;DR: In this article, the role of nonlinear dissipation on the universal escape oscillator was analyzed and its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed.
Abstract: This paper analyzes the role of nonlinear dissipation on the universal escape oscillator. Nonlinear damping terms proportional to the power of the velocity are assumed and an investigation on its effects on the dynamics of the oscillator, such as the threshold of period-doubling bifurcation, fractal basin boundaries and how the basins of attraction are destroyed, is carried out. The results suggest that increasing the power of the nonlinear damping, has similar effects as of decreasing the damping coefficient for a linearly damped case, showing the very importance of the level or amount of energy dissipation.

Journal ArticleDOI
TL;DR: In this article, a system of non-autonomous differential equations having Chua's piecewise linearity is studied, and the evolution of the dynamics and a mechanism for the development of multispiral strange attractors are discussed.
Abstract: A system of nonautonomous differential equations having Chua's piecewise-linearity is studied. A brief discussion about equilibrium points and their stability is given. It is also extended to obtain a system showing "multispiral" strange attractors, and some of the fundamental routes to "multispiral chaos" and bifurcation phenomena are demonstrated with various examples. The same work is done for other systems of autonomous or nonautonomous differential equations. This is achieved by modifying Chua's piecewise-linearity in order to have additional segments. The evolution of the dynamics and a mechanism for the development of multispiral strange attractors are discussed.

Journal ArticleDOI
TL;DR: A method for synchronizing high dimensional chaotic systems is developed to generate a linear error dynamics between the master and the slave systems so that synchronization is achievable by exploiting the controllability property of linear systems.
Abstract: In this paper a method for synchronizing high dimensional chaotic systems is developed. The objective is to generate a linear error dynamics between the master and the slave systems, so that synchronization is achievable by exploiting the controllability property of linear systems. The suggested approach is applied to Cellular Neural Networks (CNNs), which can be considered as a tool for generating complex hyperchaotic behaviors. Numerical simulations are carried out for synchronizing CNNs constituted by Chua's circuits.

Journal ArticleDOI
TL;DR: In this article, several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past.
Abstract: Several entropy-like invariants have been defined for noninvertible maps, based on various ways of measuring the dispersion of preimages and preimage sets in the past. We investigate basic properties of four such invariants, finding that their behavior in some ways differs sharply from the analogous behavior for topological entropy.

Journal ArticleDOI
TL;DR: The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches as mentioned in this paper.
Abstract: The normal forms of Hopf and generalized Hopf bifurcations have been extensively studied, and obtained using the method of normal form theory and many other different approaches It is well known that if the normal forms of Hopf and generalized Hopf bifurcations are expressed in polar coordinates, then all odd order terms must, in general, remain in the normal form In this paper, three theorems are presented to show that the conventional normal forms of Hopf and generalized Hopf bifurcations can be further simplified The forms obtained in this paper for Hopf and generalized Hopf bifurcations are shown indeed to be the "simplest", and at most only two terms remain in the amplitude equation of the "simplest normal form" up to any order An example is given to illustrate the applicability of the theory A computer algebra system using Maple is used to derive all the formulas and verify the results presented in this paper

Journal ArticleDOI
TL;DR: It is shown that the proposed method can produce accurate models which exhibit qualitatively the same dynamical behavior as the observed system and are characterized by dynamical invariants which are very close to those of the original system.
Abstract: This paper develops an original approach for identifying models of chaotic systems directly from noise-corrupted data. The nonlinear functional describing the process is constructed using a new multiresolution model structure implemented with B-spline wavelet and scaling functions. Following an iterative strategy, a sequence of model sets of increasing complexity are postulated and tested until a suitable model is found. An orthogonal-forward-regression routine coupled with model validity tests is used to select parsimonious wavelet models and to measure the quality of the fit. The effectiveness of the identification procedure is demonstrated using both simulated and experimental data. It is shown that the proposed method can produce accurate models which exhibit qualitatively the same dynamical behavior as the observed system and are characterized by dynamical invariants which are very close to those of the original system.

Journal ArticleDOI
TL;DR: The complete stability of cellular neural networks with symmetric space-variant feedback template with Dirichlet boundary condition is proved via detailed analysis on the energy function.
Abstract: We consider cellular neural networks with symmetric space-variant feedback template. The complete stability is proved via detailed analysis on the energy function. The proof is presented for the two-dimensional case with Dirichlet boundary condition. It can be extended to other dimensions with minor adjustments. Modifications to the cases of Neumann and periodic boundary conditions are also mentioned.

Journal ArticleDOI
TL;DR: The method of controlling and improvement of stability of periodic orbits of vibro-impact systems is proposed, based on the feedback loop control with a time delay.
Abstract: The method of controlling and improvement of stability of periodic orbits of vibro-impact systems is proposed. This method is based on the feedback loop control with a time delay.

Journal ArticleDOI
TL;DR: In this paper, the authors studied the structure of traveling wave solutions of cellular neural networks of the advanced type and showed the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle.
Abstract: In this paper, we study the structure of traveling wave solutions of Cellular Neural Networks of the advanced type. We show the existence of monotone traveling wave, oscillating wave and eventually periodic wave solutions by using shooting method and comparison principle. In addition, we obtain the existence of periodic wave train solutions.

Journal ArticleDOI
TL;DR: This work demonstrates the possibility of making the signal detectability at the ouput exceed that at the input when noise is added to the stochastic resonator with a simple nonlinear system that is exactly tractable analytically.
Abstract: Stochastic resonance (SR) is a nonlinear effect whereby a system is able to improve, via noise addition, the detectability of a signal in noise. SR has been demonstrated with different types of systems and signals where in each case, an appropriate detectability measure is shown improvable at the output of the stochastic resonator when noise is added at its input. A complementary issue, important for practical applications of SR, is the possibility of making the signal detectability at the ouput exceed that at the input when noise is added. We demonstrate this possibility, for both periodic and aperiodic SR, with a simple nonlinear system that we show exactly tractable analytically.

Journal ArticleDOI
TL;DR: This paper describes the winning entry of the time-series prediction competition which was part of the International Workshop on Advanced Black-Box Techniques for Nonlinear Modeling, held at K. U. Leuven, Belgium on July 8–10, 1998.
Abstract: In this paper we describe the winning entry of the time-series prediction competition which was part of the International Workshop on Advanced Black-Box Techniques for Nonlinear Modeling, held at K. U. Leuven, Belgium on July 8–10, 1998. We also describe the source of the data set, a nonlinear transform of a 5-scroll generalized Chua's circuit. Participants were given 2000 data points and were asked to predict the next 200 points in the series. The winning entry exploited symmetry that was discovered during exploratory data analysis and a method of local modeling designed specifically for the prediction of chaotic time-series. This method includes an exponentially weighted metric, a nearest trajectory algorithm, integrated local averaging, and a novel multistep ahead cross-validation estimation of model error for the purpose of parameter optimization.

Journal ArticleDOI
TL;DR: In this paper, continuous and orthogonal wavelet transforms are used to analyze time-series data and the analysis involves signal decomposition into scale components using both Grossman-Morlet and Daubechies type wavelets.
Abstract: The continuous and orthogonal wavelet transforms are used to analyze time-series data. The analysis involves signal decomposition into scale components using both Grossman–Morlet and Daubechies type wavelets. A number of simulated and experimental data vectors exhibiting different types of coherent structures, chaos and noise is analyzed. The study shows that wavelet analysis provides a unifying framework for the description of many phenomena in time-series.