Showing papers in "Chaos Solitons & Fractals in 2003"
TL;DR: In this article, an adaptive observer-based response system is designed to synchronize with a given chaotic drive system whose dynamical model is subjected to unknown parameters using the Lyapunov stability theory.
Abstract: This paper deals with the problem of synchronization of a class of continuous-time chaotic systems using the drive-response concept. An adaptive observer-based response system is designed to synchronize with a given chaotic drive system whose dynamical model is subjected to unknown parameters. Using the Lyapunov stability theory an adaptation law is derived to estimate the unknown parameters. We show that synchronization is achieved asymptotically. The approach is next applied to chaos-based secure communication. To demonstrate the efficiency of the proposed scheme numerical simulations are presented.
500 citations
TL;DR: In this paper, the authors numerically investigate chaotic behavior in autonomous nonlinear models of fractional order and show that chaotic attractors can be obtained with system orders as low as 2.1.
Abstract: We numerically investigate chaotic behavior in autonomous nonlinear models of fractional order. Linear transfer function approximations of the fractional integrator block are calculated for a set of fractional orders in ð0; 1� , based on frequency domain arguments, and the resulting equivalent models are studied. Two chaotic models are considered in this study; an electronic chaotic oscillator, and a mechanical chaotic ‘‘jerk’’ model. In both models, numerical simulations are used to demonstrate that for different types of model nonlinearities, and using the proper control parameters, chaotic attractors are obtained with system orders as low as 2.1. Consequently, we present a conjecture that third-order chaotic nonlinear systems can still produce chaotic behavior with a total system order of 2 þ e ,1 > e > 0, using the appropriate control parameters. The effect of fractional order on the chaotic range of the control parameters is studied. It is demonstrated that as the order is decreased, the chaotic range of the control parameter is affected by contraction and translation. Robustness against model order reduction is demonstrated. 2002 Elsevier Science Ltd. All rights reserved.
453 citations
TL;DR: In this article, the authors developed the Holling type II Lotka-Volterra predator-prey system, which may inherently oscillate, by introducing periodic constant impulsive immigration of predator.
Abstract: This paper develops the Holling type II Lotka–Volterra predator–prey system, which may inherently oscillate, by introducing periodic constant impulsive immigration of predator. Condition for the system to be extinct is given and permanence condition is established via the method of comparison involving multiple Liapunov functions. Further influences of the impulsive perturbations on the inherent oscillation are studied numerically, which shows that with the increasing of the amount of the immigration, the system experiences process of quasi-periodic oscillating→cycles→periodic doubling cascade→chaos→periodic halfing cascade→cycles, which is characterized by (1) quasi-periodic oscillating, (2) period doubling, (3) period halfing, (4) non-unique dynamics, meaning that several attractors coexist.
348 citations
TL;DR: In this article, the extended Jacobi elliptic function expansion method was further improved to be a more powerful method, which is still called the Extended Jacobi Elliptic Function Expansion method, by using 12 Jacobi functions.
Abstract: Our extended Jacobi elliptic function expansion method is further improved to be a more powerful method, which is still called the extended Jacobi elliptic function expansion method, by using 12 Jacobi elliptic functions. The new (2+1)-dimensional integrable Davey–Stewartson-type is chosen to illustrate the approach. As a consequence, 24 families of Jacobi elliptic function solutions are obtained. When the modulus m→1, these doubly periodic solutions degenerate as soliton solutions. The method can be also applied to other nonlinear differential equations.
266 citations
TL;DR: In this paper, a unified algebraic method was proposed to construct a series of explicit exact solutions for general nonlinear equations, including polynomial solutions, exponential solutions, rational solutions, triangular periodic wave solutions, hyperbolic, and soliton solutions, Jacobi, and Weierstrass doubly periodic wave solution.
Abstract: In this paper, we devise a new unified algebraic method to construct a series of explicit exact solutions for general nonlinear equations. Compared with most existing methods such as tanh method, Jacobi elliptic function method and homogeneous balance method, the proposed method not only gives new and more general solutions, but also provides a guideline to classify the various types of the solutions according to the values of some parameters. The solutions obtained in this paper include (a) polynomial solutions, (b) exponential solutions, (c) rational solutions, (d) triangular periodic wave solutions, (e) hyperbolic, and soliton solutions, (f) Jacobi, and Weierstrass doubly periodic wave solutions. The efficiency of the method can be demonstrated on a large variety of nonlinear equations such as those considered in this paper, combined KdV–MKdV, Camassa–Holm, Kaup–Kupershmidt, Jaulent–Miodek, (2+1)-dimensional dispersive long wave, new (2+1)-dimensional generalized Hirota, (2+1)-dimensional breaking soliton and double sine-Gordon equations. In addition, the links among our proposed method, the tanh method, the extended method and the Jacobi function expansion method are also clarified generally.
258 citations
TL;DR: A nonlinear feedback controller is designed to make the controlled system be stabilized at origin and two Genesio systems be synchronized and the stability analysis of controlled system becomes simple Hurwitz stability analysis provided that a parameter is chosen suitably.
Abstract: In this paper, a new method to control and synchronize chaotic Genesio system is proposed. We can design a nonlinear feedback controller to make the controlled system be stabilized at origin and two Genesio systems be synchronized. The stability analysis of controlled system becomes simple Hurwitz stability analysis provided that a parameter is chosen suitably. Numerical simulations verify the effectiveness of this method.
255 citations
TL;DR: Backstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller for synchronizing chaotic systems and it presents a systematic procedure for selecting a proper controller in chaos synchronization.
Abstract: Backstepping design is a recursive procedure that combines the choice of a Lyapunov function with the design of a controller. In this paper it is proposed for synchronizing chaotic systems. There are several advantages in this method for synchronizing chaotic systems: (a) it presents a systematic procedure for selecting a proper controller in chaos synchronization; (b) it can be applied to a variety of chaotic systems whether they contain external excitation or not; (c) it needs only one controller to realize synchronization between chaotic systems; (d) there is no derivatives in controller, so it is easy to be complemented. Examples of Lorenz system, Chua’s circuit and Duffing system are presented.
245 citations
TL;DR: In this paper, an effective observer is designed to identify the unknown parameter of Lu system, then the backstepping method is applied to control the uncertain Lu system to bounded points.
Abstract: Backstepping design is proposed for controlling uncertain Lu system based on parameter identification. Firstly, an effective observer is designed to identify the unknown parameter of Lu system, then the backstepping method is applied to control the uncertain Lu system to bounded points. Furthermore, it can track any continuous or discrete target. Especially, the control law designed here avoids the divergence of 1/ x and 1/ x 2 in Ref. [Chaos, Solitons & Fractals 15 (2003) 897]. Finally numerical simulations are provided to show the effectiveness and feasibility of the developed design method.
242 citations
TL;DR: In this article, a Backlund transformation both in bilinear and in ordinary form for the transformed generalised Vakhnenko equation (GVE) is derived, and an inverse scattering problem is formulated; it has a third-order eigenvalue problem.
Abstract: A Backlund transformation both in bilinear and in ordinary form for the transformed generalised Vakhnenko equation (GVE) is derived. It is shown that the equation has an infinite sequence of conservation laws. An inverse scattering problem is formulated; it has a third-order eigenvalue problem. A procedure for finding the exact N-soliton solution to the GVE via the inverse scattering method is described. The procedure is illustrated by considering the cases N=1 and 2.
232 citations
TL;DR: In this paper, a generalized method called the generally projective Riccati equation method is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccaci equation.
Abstract: A generalized method, which is called the generally projective Riccati equation method, is presented to find more exact solutions of nonlinear differential equations based upon a coupled Riccati equation. As an application of the method, we choose the higher-order nonlinear Schrodinger equation to illustrate the method. As a result more new exact travelling wave solutions are found which include bright soliton solutions, dark soliton solution, new solitary waves, periodic solutions and rational solutions. The new method can be extended to other nonlinear differential equations in mathematical physics.
191 citations
TL;DR: Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling as mentioned in this paper.
Abstract: Based on the Lyapunov stabilization theory and Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled chaotic systems with a unidirectional linear error feedback coupling. This simple criterion is applicable to a large class of chaotic systems, where only a few algebraic inequalities are involved. To demonstrate the efficiency of design, the suggested approach is applied to some typical chaotic systems with different types of nonlinearities, such as the original Chua’s circuit, the modified Chua’s circuit with a sine function, and the R€ o chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.
TL;DR: In this article, the authors considered the generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations and proved the convergence of Adomian decomposition method applied to the generalized RLW and KdV equations.
Abstract: We consider solitary-wave solutions of the generalized regularized long-wave (RLW) and Korteweg-de Vries (KdV) equations. We prove the convergence of Adomian decomposition method applied to the generalized RLW and KdV equations. Then we obtain the exact solitary-wave solutions and numerical solutions of the generalized RLW and KdV equations for the initial conditions. The numerical solutions are compared with the known analytical solutions. Their remarkable accuracy are finally demonstrated for the generalized RLW and KdV equations.
TL;DR: Lu attractor is a new chaotic attractor which connects the Lorenz attractor and Chen attractor, and represents the transition from one to the other as mentioned in this paper, and an effective observer is produced to identify the unknown parameters of Lu system.
Abstract: Lu attractor is a new chaotic attractor, which connects the Lorenz attractor and Chen attractor and represents the transition from one to the other. An effective observer is produced to identify the unknown parameters of Lu system. Moreover, a linear feedback control strategy is proposed for controlling uncertain Lu system. Numerical simulations show the effectiveness and feasibility of the proposed controllers.
TL;DR: In this article, the authors used Adomian's decomposition method for solving linear and nonlinear Klein-Gordon and sine-Gordon equations, and the obtained results show improvements over existing techniques.
Abstract: In this paper we use Adomian’s decomposition method for solving linear and nonlinear Klein–Gordon and sine-Gordon equations. Analytic and numerical studies are presented. The obtained results show improvements over existing techniques.
TL;DR: In this article, the authors address the problem of chaos control for autonomous nonlinear chaotic systems and use the recursive backstepping method of nonlinear control design to derive the nonlinear controllers, which stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction.
Abstract: In this paper, we address the problem of chaos control for autonomous nonlinear chaotic systems. We use the recursive “backstepping” method of nonlinear control design to derive the nonlinear controllers. The controller effect is to stabilize the output chaotic trajectory by driving it to the nearest equilibrium point in the basin of attraction. We study two nonlinear chaotic systems: an electronic chaotic oscillator model, and a mechanical chaotic “jerk” model. We demonstrate the robustness of the derived controllers against system order reduction arising from the use of fractional integrators in the system models. Our results are validated via numerical simulations.
TL;DR: In this paper, two chaotic Colpitts oscillators, either identical or non-identical ones, are coupled by means of two linear resistors Rk to avoid mutual synchronisation of the individual oscillators.
Abstract: The paper suggests a simple solution of building a hyperchaotic oscillator. Two chaotic Colpitts oscillators, either identical or non-identical ones are coupled by means of two linear resistors Rk. The hyperchaotic output signal v(t) is a linear combination, specifically the mean of the individual chaotic signals, v(t)=(v1+v2)/2. The corresponding differential equations have been derived. The results of both, numerical simulations and hardware experiments are presented. The coupling coefficient k∝1/Rk should be small to avoid mutual synchronisation of the individual oscillators. The spectrum of the Lyapunov exponents (LE) have been calculated versus the coefficient k. For weakly coupled oscillators there are two positive LE indicating hyperchaotic behaviour of the overall system.
TL;DR: In this paper, two different methods of control, feedback and non-feedback methods are used to suppress chaos to unstable equilibria or unstable periodic orbits (UPO) in Chen chaotic dynamical system.
Abstract: This paper is devoted to study the problem of controlling chaos in Chen chaotic dynamical system. Two different methods of control, feedback and nonfeedback methods are used to suppress chaos to unstable equilibria or unstable periodic orbits (UPO). The Lyapunov direct method and Routh–Hurwitz criteria are used to study the conditions of the asymptotic stability of the steady states of the controlled system. Numerical simulations are presented to show these results.
TL;DR: In this article, it was shown that f transitive implies f-transitive, and that f is not necessarily transitive, even if f is transitive in the Hausdorff metric space.
Abstract: Let (X,d) be a metric space and f:X→X is a continuous function. If we consider the space ( K (X),H) of all non-empty compact subsets of X endowed with the Hausdorff metric induced by d and f : K (X)→ K (X) , f (A)={f(a)/a∈A} , then the aim of this work is to show that f transitive implies f transitive. Also, we give an example showing that f transitive does not implies f transitive.
TL;DR: In this article, a new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations, including soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions.
Abstract: A new algebraic method is devised to uniformly construct a series of new travelling wave solutions for two variant Boussinesq equations. The solutions obtained in this paper include soliton solutions, rational solutions, triangular periodic solutions, Jacobi and Weierstrass doubly periodic wave solutions. Among them, the Jacobi elliptic periodic wave solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with most existing tanh methods, the proposed method gives new and more general solutions. More importantly, the method provides a guideline to classify the various types of the solution according to some parameters.
TL;DR: Firstly an observer is designed to identify the unknown parameter of Lu system, then a backstepping design method is used to control the system, and track any desired trajectory by the same way.
Abstract: In this paper, we discussed how to control Lu system with unknown parameters. Firstly we designed an observer to identify the unknown parameter of Lu system, then we used backstepping design method to control the system, and track any desired trajectory by the same way. At the same time we gave the numerical simulation for the results we had gained.
TL;DR: Using finite time control techniques, continuous state feedback control laws are developed and demonstrated that these two chaotic systems can be synchronized in finite time.
Abstract: Using finite time control techniques, continuous state feedback control laws are developed to solve the synchronization problem of two chaotic systems. We demonstrate that these two chaotic systems can be synchronized in finite time. Examples of Duffing systems, Lorenz systems are presented to verify our method.
TL;DR: In this article, a hybrid control strategy was proposed to control the period-doubling bifurcations and stabilize unstable periodic orbits embedded in the chaotic attractor of a discrete chaotic dynamical system.
Abstract: It is a typical route to generate chaos via period-doubling bifurcations in some nonlinear systems In this paper, we propose a new hybrid control strategy in which state feedback and parameter perturbation are used to control the period-doubling bifurcations and to stabilize unstable periodic orbits embedded in the chaotic attractor of a discrete chaotic dynamical system Simulation shows that the higher stable 2n-periodic orbit of the system can be controlled to lower stable 2m-periodic orbits (m
TL;DR: An adaptive feedback controller for a class of chaotic systems using Lyapunov approach to tune the controller gain vector in order to track a predetermined linearizing feedback control.
Abstract: This paper proposes an adaptive feedback controller for a class of chaotic systems. This controller can be used for tracking a smooth orbit that can be a limit cycle or a chaotic orbit of another system. Based on Lyapunov approach, the adaptation law is determined to tune the controller gain vector in order to track a predetermined linearizing feedback control. To demonstrate the efficiency of the proposed scheme, two well-known chaotic systems namely Chuas circuit and a Lure-like system are considered as illustrative examples. 2002 Elsevier Science Ltd. All rights reserved.
TL;DR: In this article, the generalized compound KdV-type and compound kdV -Burgers-type equations with nonlinear terms of any order were considered, and many explicit exact exact solutions were obtained.
Abstract: In this paper, we improved a method presented previously (Phys. Lett. A 285 (2001) 355) by means of a proper transformation. Applying the improved method, we consider the generalized compound KdV-type and compound KdV–Burgers-type equations with nonlinear terms of any order. As a result, many explicit exact solutions, which contain new solitary wave solutions, periodic wave solutions and the combined formal solitary wave solutions, are obtained.
TL;DR: In this approach a message is encrypted at the transmitter using chaotic modulation, the driving signal synchronizes the receiver using discrete observer design or drive-response concept and by reverting the coding procedure the transmitted message is reconstructed.
Abstract: In this paper we propose some secure digital communication schemes using discrete chaotic systems. In our approach a message is encrypted at the transmitter using chaotic modulation. Next, the driving signal synchronizes the receiver using discrete observer design or drive-response concept. Finally, by reverting the coding procedure the transmitted message is reconstructed. To demonstrate the efficiency of our communication schemes a modified Henon’s map is considered as an illustrative example.
TL;DR: In this article, different types of bursting Ca 2+ oscillations are classified into several subtypes based on the dynamics of separated fast and slow subsystems, the so-called fast-slow burster analysis.
Abstract: In the paper different types of bursting Ca 2+ oscillations are presented. We analyse bursting behaviour in four recent mathematical models for Ca 2+ oscillations in non-excitable cells. Separately, regular, quasi-periodic, and chaotic bursting Ca 2+ oscillations are classified into several subtypes. The classification is based on the dynamics of separated fast and slow subsystems, the so-called fast–slow burster analysis. For regular bursting Ca 2+ oscillations two types of bursting are specified: Point–Point and Point–Cycle bursting. In particular, the slow passage effect, important for the Hopf–Hopf and SubHopf–SubHopf bursting subtypes, is explained by local divergence calculated for the fast subsystem. Quasi-periodic bursting Ca 2+ oscillations can be found in only one of the four studied mathematical models and appear via a homoclinic bifurcation with a homoclinic torus structure. For chaotic bursting Ca 2+ oscillations, we found that bursting patterns resulting from the period doubling root to chaos considerably differ from those appearing via intermittency and have to be treated separately. The analysis and classification of different types of bursting Ca 2+ oscillations provides better insight into mechanisms of complex intra- and intercellular Ca 2+ signalling. This improves our understanding of several important biological phenomena in cellular signalling like complex frequency–amplitude signal encoding and synchronisation of intercellular signal transduction between coupled cells in tissue.
TL;DR: In this article, an extended Jacobian elliptic function expansion method was used to construct more exact doubly periodic solutions of the generalized Hirota-Satsuma coupled KdV system by using symbolic computation.
Abstract: In this paper an extended Jacobian elliptic function expansion method, which is a direct and more powerful method, is used to construct more new exact doubly periodic solutions of the generalized Hirota–Satsuma coupled KdV system by using symbolic computation. As a result, sixteen families of new doubly periodic solutions are obtained which shows that the method is more powerful. When the modulus of the Jacobian elliptic functions m →1 or 0, the corresponding six solitary wave solutions and six trigonometric function (singly periodic) solutions are also found. The method is also applied to other higher-dimensional nonlinear evolution equations in mathematical physics.
TL;DR: The feed forward neural networks (FFNN) is used to model the charged multiplicity distribution of K–P interactions at high energies and results showed good fitting to the experimental data.
Abstract: Artificial intelligence techniques involving neural networks became vital modeling tools where model dynamics are difficult to track with conventional techniques. The paper make use of the feed forward neural networks (FFNN) to model the charged multiplicity distribution of K–P interactions at high energies. The FFNN was trained using experimental data for the multiplicity distributions at different lab momenta. Results of the FFNN model were compared to that generated using the parton two fireball model and the experimental data. The proposed FFNN model results showed good fitting to the experimental data. The neural network model performance was also tested at non-trained space and was found to be in good agreement with the experimental data.
TL;DR: In this paper, Pascual-Leone obtained a fit between predicted and tested span by applying Bose-Einstein-statistics to learning experiments, and Multiplying span by mental speed and using the entropy formula for bosons, they obtained the same result.
Abstract: The principle of information coding by the brain seems to be based on the golden mean. For decades psychologists have claimed memory span to be the missing link between psychometric intelligence and cognition. By applying Bose–Einstein-statistics to learning experiments, Pascual-Leone obtained a fit between predicted and tested span. Multiplying span by mental speed (bits processed per unit time) and using the entropy formula for bosons, we obtain the same result. If we understand span as the quantum number n of a harmonic oscillator, we obtain this result from the EEG. The metric of brain waves can always be understood as a superposition of n harmonics times 2Φ, where half of the fundamental is the golden mean Φ (=1.618) as the point of resonance. Such wave packets scaled in powers of the golden mean have to be understood as numbers with directions, where bifurcations occur at the edge of chaos, i.e. 2Φ=3+φ3. Similarities with El Naschie’s theory for high energy particle’s physics are also discussed.
TL;DR: In this paper, a framework of calculus on net fractals is built, where integrals and derivatives of functions in net measure are discussed and applications of the calculus in some physical systems, such as in diffusion processes and in memory processes are given.
Abstract: In this paper a framework of calculus on net fractals is built. Integrals and derivatives of functions in net measure are discussed. Approximate calculations of the integrals and derivatives and approximate solutions to the inverse problem of integrals in net measure are given. In addition, applications of the calculus in some physical systems, such as in diffusion processes and in memory processes are given.