Large Class Of Chaotic Systems Research Articles

Robust observer-based synchronisation of chaotic oscillators with structural perturbations and input nonlinearity

This paper presents a generalised robust adaptive chaotic synchronisation method for chaotic systems with structural perturbations. One control input is used to synchronise both systems exponentially fast based on Lyapunov theory. This approach cannot only make the outputs of both master and slave systems reach synchronisation with the passage of time between both systems but it can also reduce the effect of external perturbations and input nonlinearities. By assuming bounded solutions of the nominal uncoupled systems, sufficient conditions have been derived for boundedness of the solutions of two different classes of chaotic systems with input nonlinearity affected by structural perturbations. The propose approach offers a systematic design procedure for robust adaptive synchronisation of a large class of chaotic systems in the chaos research literature. As an illustration of the effectiveness and robustness of the proposed strategy, synchronisation problem of a master system consists of a perturbed modified Colpitts oscillator and an observer consisting of a Chua oscillator. It was found that the controller maintains robust stable synchronisation in the presence of exoteric perturbations and structural uncertainties.

Symmetry and dynamics universality of supermetal in quantum chaos

Chaotic systems exhibit rich quantum dynamical behaviors ranging from dynamical localization to normal diffusion to ballistic motion. Dynamical localization and normal diffusion simulate electron motion in an impure crystal with a vanishing and finite conductivity, i.e., an "Anderson insulator" and a "metal", respectively. Ballistic motion simulates a perfect crystal with diverging conductivity, i.e., a "supermetal". We analytically find and numerically confirm that, for a large class of chaotic systems, the metal-supermetal dynamics crossover occurs and is universal, determined only by the system's symmetry. Furthermore, we show that the universality of this dynamics crossover is identical to that of eigenfunction and spectral fluctuations described by the random matrix theory.

Adaptive Control for Modified Projective Synchronization-Based Approach for Estimating All Parameters of a Class of Uncertain Systems: Case of Modified Colpitts Oscillators

A method of estimation of all parameters of a class of nonlinear uncertain dynamical systems is considered, based on the modified projective synchronization (MPS). The case of modified Colpitts oscillators is investigated. Through a suitable transformation of the dynamical system, sufficient conditions for achieving synchronization are derived based on Lyapunov stability theory. Global stability and asymptotic robust synchronization of the considered system are investigated. The proposed approach offers a systematic design procedure for robust adaptive synchronization of a large class of chaotic systems. The combined effect of both an additive white Gaussian noise (AWGN) and an artificial perturbation is numerically investigated. Results of numerical simulations confirm the effectiveness of the proposed control strategy.

Sampled-data nonlinear observer design for chaos synchronization: A Lyapunov-based approach

This paper considers sampled-data based chaos synchronization using observers in the presence of measurement noise for a large class of chaotic systems. We study discretized model of chaotic systems which are perturbed by white noise and employ Lyapunov-like theorems to come up with a simple yet effective observer design. For the choice of observer gain, a suboptimal criterion is obtained in terms of LMI. We present semiglobal as well as global results. The proposed scheme can also be extended for discrete-time chaotic systems. Numerical simulations have been carried out to verify the effectiveness of theoretical results.

Synchronization of Multi-Chaotic Systems with Ring and Chain Intermittent Connections

Based on the intermittent control and the nonlinear measure about l 2-norm, a novel and effective approach to synchronization of multi-chaotic systems with ring and chain connections is investigated. The proposed approach offers a design procedure for multi-chaos synchronization of a large class of chaotic systems. Numerical simulations of multi-chaotic Lorenz systems are given to illustrate the main results by choosing appropriate coupling coefficients by using the intermittent controllers.

Sum-of-square-based impulsive control for chaotic system based on polynomial model

In this paper, we present a novel impulsive control method based on polynomial model for a large class of chaotic systems. First, the polynomial model is used to model the chaotic system, in which the state equation of the system is composed of the polynomial matrix of the system and the column vector of monomials in state. Compared with others modeling methods, any pre-defined hypothesis is removed. Next, a sum-of-square (SOS)-based impulsive control method is investigated to guarantee that the chaotic system is asymptotically stable. It can obtain larger impulsive interval using SOS-based optimization algorithm over linear matrix inequality technique, which means the same control performance can be realized by less control action. Finally, the simulation is provided to demonstrate the effectiveness of the proposed method.

Simple adaptive variable structure control for unknown chaotic systems

This paper proposes two novel adaptive variable structure tracking controllers for a large class of chaotic systems with unknown dynamics in presence of both external disturbances and input nonlinearities. The pros and cons of each proposed methodology is also represented. In order to eliminate the chattering effect in the former controlled system, two corresponding fuzzy adaptive controllers are presented. Besides, synchronization of two non-identical uncertain chaotic systems is investigated using our proposed methods in both full and reduced-order forms. It can be seen that not only our proposed control schemes can be applied to a wide class of uncertain chaotic systems but also it is simple to implement in practical application. Finally, the proposed methods are applied to some famous chaotic systems to verify the effectiveness of the proposed methods.

Controlling chaos using Takagi–Sugeno fuzzy model and adaptive adjustment

In this paper, an approach to the control of continuous-time chaotic systems is proposed using the Takagi–Sugeno (TS) fuzzy model and adaptive adjustment. Sufficient conditions are derived to guarantee chaos control from Lyapunov stability theory. The proposed approach offers a systematic design procedure for stabilizing a large class of chaotic systems in the literature about chaos research. The simulation results on Rossler's system verify the effectiveness of the proposed methods.

SYNCHRONIZATION BY OBSERVER BASED APPROACH FOR A CLASS OF NONLINEAR CHAOTIC SYSTEMS

In this paper, we present a new approach based observers for the synchronization of a large class of chaotic systems. Thanks to the Lyapunov theory we give new sufficient synthesis conditions for the synchronization problem. These conditions are formulated as Linear Matrix Inequalities (LMIs) which are easily tractable using standards optimization algorithms. Numerical examples are provided to show high performances of the proposed approach and the large class of chaotic systems that are concerned.

A new output feedback synchronization theorem for a class of chaotic systems with a scalar transmitted signal

This paper proposes a new, simple and yet applicable output feedback synchronization theorem for a large class of chaotic systems. We take a linear combination of drive system state variables as a scale-driving signal. It is proved that synchronization between the drive and the response systems can be obtained via a simple linear output error feedback control. The linear feedback gain is a function of a free parameter. The approach is illustrated using the Rössler hyperchaotic systems and Chua's chaotic oscillators.

Two novel synchronization criterions for a unified chaotic system

Two novel synchronization criterions are proposed in this paper. It includes drive–response synchronization and adaptive synchronization schemes. Moreover, these synchronization criterions can be applied to a large class of chaotic systems and are very useful for secure communication.

A NEW APPROACH FOR THE OBSERVER-BASED SYNCHRONIZATION OF CHAOTIC SYSTEMS

In this paper, a new framework for the synchronization of chaotic systems is presented. The synchronization problem of a large class of chaotic systems is formulated as an observer synthesis problem for an appropriate class of linear parameter-varying (LPV) systems. The result introduced in this paper shows that LPV techniques can successfully be used in the context of chaotic systems synchronization. Two examples are considered in order to show the applicability of the LPV approach.

Secure synchronization of a class of chaotic systems from a nonlinear observer approach

The aim of this note is two-fold. First, it discusses synchronization of chaotic systems from a control theoretic point of view and introduces the concept of secure synchronization, i.e., a communication scheme that resists possible intrusion based on either adaptive or robust control techniques. Second, for a large class of chaotic systems, i.e., the generalized Lorenz system, global exponential synchronization via a scalar communication signal is suggested and its security is analyzed from a control theoretic viewpoint. Both theoretical analysis and numerical simulations are provided, verifying the proposed chaos-synchronization-based secure communication design principle and methodology.

Global synchronization criterion and adaptive synchronization for new chaotic system

Abstract This paper proposes two schemes of synchronization of two four-scorll chaotic attractor, a simple global synchronization and adaptive synchronization in the presence of unknown system parameters. Based on Lyapunov stability theory and matrix measure, a simple generic criterion is derived for global synchronization of four-scorll chaotic attractor system with a unidirectional linear error feedback coupling. This methods are applicable to a large class of chaotic systems where only a few algebraic inequalities are involved. Numerical simulations are presented to show the effectiveness of the proposed chaos synchronization method.

CHAOS SYNCHRONIZATION OF TWO COUPLED DYNAMOS SYSTEMS WITH UNKNOWN SYSTEM PARAMETERS

This paper addresses the synchronization problem of two coupled dynamos systems in the presence of unknown system parameters. Based on Lyapunov stability theory, an active control law is derived and activated to achieve the state synchronization of two identical coupled dynamos systems. By using Gerschgorin theorem, a simple generic criterion is derived for global synchronization of two coupled dynamos 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. Numerical simulations results are used to demonstrate the effectiveness of the proposed control methods.

Synchronization-based estimation of all parameters of chaotic systems from time series.

By a simple combination of adaptive scheme and linear feedback with the updated feedback strength, for a large class of chaotic systems it is proved rigorously by using the invariance principle of differential equations that all unknown model parameters can be estimated dynamically. This approach supplies a systematic and analytical procedure for estimating parameters from time series, and it is simple to implement in practice. In addition, this method is quite robust against the effect of noise and able to respond rapidly to changes in operating parameters of the experimental system. Lorenz and Rössler hyperchaos systems are used to illustrate the validity of this technique.

A New Criterion for Chaos Synchronization Using Linear State Feedback Control

This paper studies chaos synchronization between two coupled chaotic systems using linear state feedback control. A new, simple and yet easily applicable criterion is derived for chaos synchronization based on the Lyapunov stability theory and the Linear Quadratic Optimal Control theory. This criterion is given in terms of some simple algebraic inequalities established by the Gerschgorin theorem, and it is easily applicable to a large class of chaotic systems. As an example, the familiar Chua's circuit is simulated for demonstration.

A simple global synchronization criterion for coupled chaotic systems

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 Rössler chaotic system. It is proved that these synchronizations are ensured by suitably designing the coupling parameters.

HYPERBOLIC-TYPE GENERALIZED LORENZ CHAOTIC SYSTEM AND ITS CANONICAL FORM

This paper shows that a large class of chaotic systems, introduced in (Čelikovský and Vaněček, 1994), (Vaněček and Čelikovský, 1996) as the generalized Lorenz system, can be further generalized to the hyperbolic-type generalized Lorenz system. While the generalized Lorenz system unifies both the famous Lorenz system and new Chen's system (Ueta and Chen, 1999), (Chen and Ueta, 2000), the hyperbolic-type generalized Lorenz system introduced here is in some way complementary to it. Such a complementarity is especially clear when considering the canonical form of the generalized Lorenz system obtained in (Čelikovský and Chen, 2002), where the canonical form is characterized by the eigenvalues of the linearized part together with a key parameter τ ∈ (–1, ∞). The analogous canonical form of the hyperbolic-type generalized Lorenz system introduced here corresponds to the case of, while τ = –1 is a single special case. This new class of chaotic systems is then analyzed, both analytically and numerically, showing its rich variety of dynamical behaviours, including bifurcation and chaos. Moreover, an algorithm for transforming the hyberbolic-type generalized Lorenz system into its canonical form, as well as its inverse scheme, are presented.

Stabilizing nonlinear dynamical systems by an adaptive adjustment mechanism

An adaptive adjustment mechanism is applied to stabilize multidimensional dynamical systems. Without utilizing any prior knowledge of the system itself, nor extra external control signals, the mechanism can ensure a large class of chaotic systems to converge to their "generic" stable periodic orbits.