Showing papers in "International Journal of Bifurcation and Chaos in 2002"
TL;DR: This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractsor and represents the transition from one to the other.
Abstract: This letter reports the finding of a new chaotic attractor in a simple three-dimensional autonomous system, which connects the Lorenz attractor and Chen's attractor and represents the transition from one to the other.
1,655 citations
TL;DR: It is shown that, for any given coupling strength and a suciently large number of cells, the small-world dynamical network will synchronize, even if the original nearest-neighbor coupled network cannot achieve synchronization under the same condition.
Abstract: We investigate synchronization in a network of continuous-time dynamical systems with smallworld connections. The small-world network is obtained by randomly adding a small fraction of connection in an originally nearest-neighbor coupled network. We show that, for any given coupling strength and a suciently large number of cells, the small-world dynamical network will synchronize, even if the original nearest-neighbor coupled network cannot achieve synchronization under the same condition.
814 citations
TL;DR: A unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum and is chaotic over the entire spectrum of the key system parameter.
Abstract: This paper introduces a unified chaotic system that contains the Lorenz and the Chen systems as two dual systems at the two extremes of its parameter spectrum. The new system represents the continued transition from the Lorenz to the Chen system and is chaotic over the entire spectrum of the key system parameter. Dynamical behaviors of the unified system are investigated in somewhat detail.
806 citations
TL;DR: This paper reviews some important results in this direction of rapidly evolving research in the field of complex networks, with emphasis on the areas of reinforcement learning and reinforcement learning.
Abstract: Dramatic advances in the field of complex networks have been witnessed in the past few years. This paper reviews some important results in this direction of rapidly evolving research, with emphasis...
537 citations
TL;DR: The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design.
Abstract: This paper shows that a large class of systems, introduced in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996] as the so-called generalized Lorenz system, are state-equivalent to a special canonical form that covers a broader class of chaotic systems. This canonical form, called generalized Lorenz canonical form hereafter, generalizes the one introduced and analyzed in [Celikovský & Vaněcek, 1994; Vaněcek & Celikovský, 1996], and also covers the so-called Chen system, recently introduced in [Chen & Ueta, 1999; Ueta & Chen, 2000]. Thus, this new generalized Lorenz canonical form contains as special cases the original Lorenz system, the generalized Lorenz system, and the Chen system, so that a comparison of the structures between two essential types of chaotic systems becomes possible. The most important property of the new canonical form is the parametrization that has precisely a single scalar parameter useful for chaos tuning, which has promising potential in future engineering chaos design. Some...
314 citations
TL;DR: A number of strange attractors from the scroll grid attractor families are presented and have been experimentally verified using current feedback opamps to show the generalization of the nonlinear characteristics.
Abstract: In this paper a new family of scroll grid attractors is presented. These families are classified into three called 1D-, 2D- and 3D-grid scroll attractors depending on the location of the equilibrium points in state space. The scrolls generated from 1D-, 2D- and 3D-grid scroll attractors are located around the equilibrium points on a line, on a plane or in 3D, respectively. Due to the generalization of the nonlinear characteristics, it is possible to increase the number of scrolls in all state variable directions. A number of strange attractors from the scroll grid attractor families are presented. They have been experimentally verified using current feedback opamps. Also Lur'e representations are given for the scroll grid attractor families.
267 citations
TL;DR: Dynamical behaviors of a new chaotic attractor is investigated and the transition between the Lorenz attractor and Chen's attractor through the new system is explored.
Abstract: Dynamical behaviors of a new chaotic attractor is investigated in this paper. Some basic properties, bifurcations, routes to chaos, and periodic windows of the new system are studied either analytically or numerically. Meanwhile, the transition between the Lorenz attractor and Chen's attractor through the new system is explored.
238 citations
IBM1
TL;DR: A new continuation method for computing implicitly defined manifolds is presented, represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a bounda...
Abstract: We present a new continuation method for computing implicitly defined manifolds. The manifold is represented as a set of overlapping neighborhoods, and extended by an added neighborhood of a bounda...
141 citations
TL;DR: It is demonstrated that often spatial extension is indeed capable of promoting cooperative behavior and this holds in particular for the prisoner's dilemma for a small but important parameter range.
Abstract: A systematic analysis of the effects of spatial extension on the equilibrium frequency of cooperators and defectors in 2 × 2 games is presented and compared to well mixed populations where spatial extension can be neglected. We demonstrate that often spatial extension is indeed capable of promoting cooperative behavior. This holds in particular for the prisoner's dilemma for a small but important parameter range. For the hawk–dove game, spatial extension may lead to both, increases of the hawk- as well as the dove-strategy. The outcome subtly depends on the parameters as well as on the degree of stochasticity in the different update rules. For rectangular lattices, the general conclusions are rather robust and hold for different neighborhood types i.e. for the von Neumann as well as the Moore neighborhood and, in addition, they appear to be almost independent of the update rule of the lattice. However, increasing stochasticity for the update rules of the players results in equilibrium frequencies more closely related to the mean field description.
135 citations
TL;DR: The aim of this paper is to review some of the mechanisms which leads to stable localized patterns in nonequilibrium systems.
Abstract: The aim of this paper is to review some of the mechanisms which leads to stable localized patterns in nonequilibrium systems.
95 citations
TL;DR: A new circuitry design based on Chua’s circuit for generating n-scroll attractors (n =1 ; 2; 3;:::) is proposed, and chaotic attractors with an even or an odd number of scrolls can be easily generated.
Abstract: A new circuitry design based on Chua’s circuit for generating n-scroll attractors (n =1 ; 2; 3;:::) is proposed. In this design, the nonlinear resistor in Chua’s circuit is constructed via a systematical procedure using basic building blocks. With the proposed construction scheme, the slopes and break points of the v{i characteristic of the circuit can be tuned independently, and chaotic attractors with an even or an odd number of scrolls can be easily generated. Distinct attractors with n-scrolls (n =5 ; 6; 7; 8; 9; 10) obtained with this simple experimental set-up are demonstrated.
TL;DR: A new practical method for distinguishing chaotic, periodic and quasi-periodic orbits based on a new criterion is introduced, and it is applied to investigate the local bifurcations of the Chen system.
Abstract: This paper introduces a new practical method for distinguishing chaotic, periodic and quasi-periodic orbits based on a new criterion, and apply it to investigate the local bifurcations of the Chen system. Conditions for supercritical and subcritical bifurcations are obtained, with their parameter domains specified. The analytic results are also verified by numerical simulation studies.
TL;DR: This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science by mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, and shows how some ofWolfram's empirical observations can be justified on firm ground.
Abstract: This tutorial provides a nonlinear dynamics perspective to Wolfram's monumental work on A New Kind of Science. By mapping a Boolean local Rule, or truth table, onto the point attractors of a specially tailored nonlinear dynamical system, we show how some of Wolfram's empirical observations can be justified on firm ground. The advantage of this new approach for studying Cellular Automata phenomena is that it is based on concepts from nonlinear dynamics and attractors where many fuzzy concepts introduced by Wolfram via brute force observations can be defined and justified via mathematical analysis. The main result of Part I is the introduction of a fundamental concept called linear separability and a complexity index κ for each local Rule which characterizes the intrinsic geometrical structure of an induced "Boolean cube" in three-dimensional Euclidean space. In particular, Wolfram's seductive idea of a "threshold of complexity" is identified with the class of local Rules having a complexity index equal to 2.
TL;DR: Based on Lyapunov stabilization theory, a new generic criterion of global chaos synchronization between two coupled chaotic systems from a unidirectional linear error feedback coupling approach is proposed.
Abstract: Based on Lyapunov stabilization theory, this paper proposes a new generic criterion of global chaos synchronization between two coupled chaotic systems from a unidirectional linear error feedback coupling approach. The criterion is successfully applied to some typical chaotic systems with dierent types of nonlinearity, such as the classic Chua’s circuit, the modied
TL;DR: The generalized Henon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions.
Abstract: The generalized Henon maps (GHM) are discrete-time systems with given finite dimension, which show chaotic and hyperchaotic behavior for certain parameter values and initial conditions. A study of these maps is given where particularly higher-dimensional cases are considered.
TL;DR: The results show that the proposed controller can steer Chua’s circuit system to the desired state without the chattering phenomenon and abrupt state change.
Abstract: In this paper, a sliding mode controller is applied to control the cubic Chua’s circuit system. The sliding surface of this paper used is one dimension higher than the traditional surface and guarantees its passage through the initial states of the controlled system. Therefore, using the characteristic of this sliding mode we aim to design a controller that can meet the desired specication and use less control energy by comparing with the result in the current existing literature. The results show that the proposed controller can steer Chua’s circuit system to the desired state without the chattering phenomenon and abrupt state change.
TL;DR: It is shown that PS could, by means of control, be extended to general classes of chaotic systems with nonpartial linearity, and the feasibility of the control for high dimensional systems is demonstrated in a hyperchaotic system.
Abstract: Projective synchronization (PS), in which the state vectors synchronize up to a scaling factor, is usually observable only in partially linear systems. We show that PS could, by means of control, be extended to general classes of chaotic systems with nonpartial linearity. Performance of PS may also be manipulated by controlling the scaling factor to any desired value. In numerical experiments, we illustrate the applications to a Rossler system and a Chua's circuit. The feasibility of the control for high dimensional systems is demonstrated in a hyperchaotic system.
TL;DR: This Letter presents an experimental realization of a recently proposed method to anticipate future states of nonlinear time-delayed feedback systems and finds that synchronization of the driven circuit with chaotic futureStates of the driving circuit is insensitive to signal and system perturbations.
Abstract: This Letter presents an experimental realization of a recently proposed method to anticipate future states of nonlinear time-delayed feedback systems. The electronic circuit allows for a real-time anticipation of even strongly irregular signals. It is found that synchronization of the driven circuit with chaotic future states of the driving circuit is insensitive to signal and system perturbations.
TL;DR: A robust adaptive neural control scheme for a class of uncertain chaotic systems in the disturbed strict-feedback form, with both unknown nonlinearities and uncertain disturbances is presented.
Abstract: Though chaotic behaviors are exhibited in many simple nonlinear models, physical chaotic systems are much more complex and contain many types of uncertainties. This paper presents a robust adaptive neural control scheme for a class of uncertain chaotic systems in the disturbed strict-feedback form, with both unknown nonlinearities and uncertain disturbances. To cope with the two types of uncertainties, we combine backstepping methodology with adaptive neural design and nonlinear damping techniques. A smooth singularity-free adaptive neural controller is presented, where nonlinear damping terms are used to counteract the disturbances. The differentiability problem in controlling the disturbed strict-feedback system is solved without employing norm operation, which is usually used in robust control design. The proposed controllers can be applied to a large class of uncertain chaotic systems in practical situations. Simulation studies are conducted to verify the effectiveness of the scheme.
TL;DR: It is proved that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions, as well as new examples of chaotic backward shift operators.
Abstract: Backward shift operators provide a general class of linear dynamical systems on infinite dimensional spaces. Despite linearity, chaos is a phenomenon that occurs within this context. In this paper we give characterizations for chaos in the sense of Auslander and Yorke [1980] and in the sense of Devaney [1989] of weighted backward shift operators and perturbations of the identity by backward shifts on a wide class of sequence spaces. We cover and unify a rich variety of known examples in different branches of applied mathematics. Moreover, we give new examples of chaotic backward shift operators. In particular we prove that the differential operator I + D is Auslander–Yorke chaotic on the most usual spaces of analytic functions.
TL;DR: The dynamical analysis of intelligent systems that use shape memory actuators involves a multi-degree of freedom system and shows some relation between the transmissibility of order, chaos and hyperchaos with temperature.
Abstract: Shape memory and pseudoelastic effects are thermomechanical phenomena associated with martensitic phase transformations, presented by shape memory alloys. The dynamical analysis of intelligent systems that use shape memory actuators involves a multi-degree of freedom system. This contribution concerns with the chaotic response of shape memory systems. Two different systems are considered: a single and a two-degree of freedom oscillator. Equations of motion are formulated assuming a polynomial constitutive model to describe the restitution force of oscillators. Since equations of motion of the two-degree of freedom oscillator are associated with a five-dimensional system, the analysis is performed considering two oscillators, both with single-degree of freedom, connected by a spring-dashpot system. With this assumption, it is possible to analyze the transmissibility of motion between two oscillators. Results show some relation between the transmissibility of order, chaos and hyperchaos with temperature.
TL;DR: This paper carries out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and shows that they are nondegenerate, which allows to put in evidence the presence of several kinds of bifurstations of periodic orbits and of global phenomena.
Abstract: This paper is devoted to the analysis of bifurcations in a three-parameter unfolding of a linear degeneracy corresponding to a triple-zero eigenvalue. We carry out the study of codimension-two local bifurcations of equilibria (Takens–Bogdanov and Hopf-zero) and show that they are nondegenerate. This allows to put in evidence the presence of several kinds of bifurcations of periodic orbits (secondary Hopf,…) and of global phenomena (homoclinic, heteroclinic). The results obtained are applied in the study of the Rossler equation.
TL;DR: An electronic circuit realization of a modified Lorenz system, which is multiplier-free, is described and it is clarified that the butterfly attractor is a compound structure obtained by merging together two simple attractors after performing one mirror operation.
Abstract: An electronic circuit realization of a modified Lorenz system, which is multiplier-free, is described. The well-known butterfly chaotic attractor is experimentally observed verifying that the proposed modified system does capture the essential dynamics of the original Lorenz system. Furthermore, we clarify that the butterfly attractor is a compound structure obtained by merging together two simple attractors after performing one mirror operation.
TL;DR: This paper realizes Chen's attractor with an electronic circuit, the first design reported, and the synchronization scheme between two Chen’s attractors is also considered and realized.
Abstract: Chen's attractor was recently reported and proved to be topologically different from the Lorenz butterfly attractor. In this paper, Chen's attractor is realized with an electronic circuit. Although the circuitry is straight-forward, it is the first design reported. In addition, the synchronization scheme between two Chen's attractors is also considered and realized.
TL;DR: The leading large-n behavior that may be expected on general grounds is established, where Nn is the number of sites on the nth generation tower and is the spectral dimension of the fractal.
Abstract: We consider an unbiased random walk on a finite, nth generation Sierpinski gasket (or "tower") in d = 3 Euclidean dimensions, in the presence of a trap at one vertex. The mean walk length (or mean number of time steps to absorption) is given by the exact formula The generalization of this formula to the case of a tower embedded in an arbitrary number d of Euclidean dimensions is also found, and is given by This also establishes the leading large-n behavior that may be expected on general grounds, where Nn is the number of sites on the nth generation tower and is the spectral dimension of the fractal.
TL;DR: It is pointed out that under the influence of a periodic excitation a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, occur in this circuit.
Abstract: We present a detailed investigation of the rich variety of bifurcations and chaos associated with a very simple nonlinear parallel nonautonomous LCR circuit with Chua's diode as its only nonlinear element as briefly reported recently [Thamilmaran et al., 2000]. It is proposed as a variant of the simplest nonlinear nonautonomous circuit introduced by Murali, Lakshmanan and Chua (MLC) [Murali et al., 1994]. In our study we have constructed two-parameter phase diagrams in the forcing amplitude-frequency plane, both numerically and experimentally. We point out that under the influence of a periodic excitation a rich variety of bifurcation phenomena, including the familiar period-doubling sequence, intermittent and quasiperiodic routes to chaos as well as period-adding sequences, occur. In addition, we have also observed that the periods of many windows satisfy the familiar Farey sequence. Further, reverse bifurcations, antimonotonicity, remerging chaotic band attractors, and so on, also occur in this system. Numerical simulation results using Poincare section, Lyapunov exponents, bifurcation diagrams and phase trajectories are found to be in agreement with experimental observations. The chaotic dynamics of this circuit is observed experimentally and confirmed both by numerical and analytical studies as well PSPICE simulation results. The results are also compared with the dynamics of the original MLC circuit with reference to the two-parameter space to show the richness of the present circuit.
TL;DR: Using Lyapunov stability theory the adaptive synchronization law with a single-state variable feedback is derived, such that the trajectory of the two systems are globally stabilized to an equilibrium point of the uncontrolled system.
Abstract: This study addresses the adaptive synchronization of Rossler and Chua circuit systems with unknown parameters. By using Lyapunov stability theory the adaptive synchronization law with a single-state variable feedback is derived, such that the trajectory of the two systems are globally stabilized to an equilibrium point of the uncontrolled system (globally stable means that the method of the solution is restricted in area of phase space i.e. globally in a subset of a phase space with bounded zero volume). We use the Lyapunov direct method to study the asymptotic stability of the solutions of error system. Numerical simulations are given to explain the effectiveness of the proposed control scheme.
TL;DR: A method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle using a finite piece of tangle that can be computed with available numer...
Abstract: We present a method for finding symbolic dynamics for a planar diffeomorphism with a homoclinic tangle. The method only requires a finite piece of tangle, which can be computed with available numer...
TL;DR: A simple continuous-time system for chaos generation based on a third-order abstract canonical mathematical model, which reflects the behavior of a digital inverter, which is introduced by a bipolar switching constant.
Abstract: We propose a simple continuous-time system for chaos generation based on a third-order abstract canonical mathematical model. Nonlinearity in this model is introduced by a bipolar switching constant, which reflects the behavior of a digital inverter. A simple area efficient implementation of the system in a 1.2 μ CMOS process is presented. Experimental results from a tested chip are shown.
TL;DR: A new impulsive control method for chaos suppression, using Chen's system as an example, is developed and new general criteria for exponential stability and asymptotical stability of nonlinear impulsive systems are established.
Abstract: In this paper, impulsive control of nonlinear systems and its application to Chen's chaotic system are considered. A new impulsive control method for chaos suppression, using Chen's system as an example, is developed. Some new general criteria for exponential stability and asymptotical stability of nonlinear impulsive systems are established and, particularly, some simple conditions sufficient for driving the chaotic state of Chen's system to its zero equilibrium are presented.