Observer-based impulsive chaotic synchronization of discrete-time switched systems

This paper investigates impulsive chaotic synchronization of discrete-time switched systems with state-dependent switching strategy. The parameter-dependent Lyapunov function (PDLF) technique is used to establish stability criteria for a class of switched systems consisting of both stable and unstable subsystems. With these criteria, sufficient conditions are given to achieve observer-based impulsive chaotic synchronization. Examples are presented to illustrate the criteria.     

Reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems

The problem of reliable impulsive lag synchronization for a class of nonlinear discrete chaotic systems is investigated in this paper. Firstly a reliable impulsive controller is designed by the impulsive control theory. Then, some sufficient conditions for reliable impulsive lag synchronization between the drive system and the response system are obtained. Numerical simulations are given to show the effectiveness of the proposed method.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Robust adaptive synchronization of uncertain unified chaotic systems

The problem of synchronizing a unified chaotic system in the presence of parameter variations, unstructured uncertainties, and external disturbances is addressed. To tackle such perturbations whose bounds may be unknown, two robust adaptive algorithms are proposed. The stability analysis is presented based on the Lyapunov stability theorem. Simulation results demonstrate the performance of the developed synchronization schemes.     

Adaptive control for synchronization of a four-dimensional chaotic system via a single variable

This paper studies the control for synchronization of a four-dimensional system via a single variable, and a linear feedback controller and an adaptive controller are proposed. Based on the Lyapunov stability theory, the correctness of the proposed methods is strictly demonstrated. The numerical simulations further show their effectiveness.     

A novel active pinning control for synchronization and anti-synchronization of new uncertain unified chaotic systems

This paper discusses the synchronization and anti-synchronization of new uncertain unified chaotic systems (UUCS). Based on the idea of active control, a novel active Pinning control strategy is presented, which only needs a state of new UUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UUCS. Numerical simulations of new UUCS show that the controller can make chaotic systems achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.     

Adaptive feedback control and synchronization of non-identical chaotic fractional order systems

This paper addresses the reliable synchronization problem between two non-identical chaotic fractional order systems. In this work, we present an adaptive feedback control scheme for the synchronization of two coupled chaotic fractional order systems with different fractional orders. Based on the stability results of linear fractional order systems and Laplace transform theory, using the master-slave synchronization scheme, sufficient conditions for chaos synchronization are derived. The designed controller ensures that fractional order chaotic oscillators that have non-identical fractional orders can be synchronized with suitable feedback controller applied to the response system. Numerical simulations are performed to assess the performance of the proposed adaptive controller in synchronizing chaotic systems.     

Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.     

Exponential synchronization of chaotic Lur’e systems with delayed feedback control

The exponential synchronization problem is studied in this paper for a class of chaotic Lur’e systems by using delayed feedback control. An augmented Lyapunov functional based approach is proposed to deal with this issue. A delay-dependent condition is established such that the controlled slave system can exponentially synchronize with the master system. It is shown that the delayed feedback gain matrix and the exponential decay rate can be obtained by solving a set of linear matrix inequalities. The decay coefficient can be also easily calculated. Finally, as an example, the Chua’s circuit is used to illustrate the effectiveness of the developed approach and the improvement over some existing results.     

Nonlinear observer-based impulsive synchronization in chaotic systems with multiple attractors

The issue of impulsive synchronization of the coupled Newton–Leipnik system is investigated. Based on the impulsive stability theory, nonlinear observer-based impulsive synchronization scheme is derived. A new and less conservative criteria for impulsive synchronization via nonlinear observer is proposed. The boundary of the stable regions is also estimated. One important advantage of the proposed method is that it is also applicable for the systems with more than one attractor. Numerical simulations on Newton–Leipnik system are illustrated to verify the theoretical results.     

New results on synchronization of chaotic systems with time-varying delays via intermittent control

This paper is concerned with the problem of exponential synchronization for chaotic systems with time-varying delays by using periodically intermittent control. Some new and useful synchronization criteria are obtained based on the differential inequality method and the analysis technique. It is noteworthy that the methods used in this paper are different from the techniques employed in the existing works, and the derived conditions are less conservative. Especially, a strong constraint on the control width that the control width should be large than the time delay imposed by the current references is released in this paper. Moreover, the new synchronization criteria do not impose any restriction on the size of time delay. Numerical examples are finally presented to illustrate the effectiveness of the theoretical results.     

Observer-based projective synchronization of fractional systems via a scalar signal: application to hyperchaotic R?ssler systems

This paper introduces an observer-based approach to achieve projective synchronization in fractional-order chaotic systems using a scalar synchronizing signal. The proposed method, which enables a linear fractional error system to be obtained, exploits the Kalman decomposition and a proper stability criterion in order to stabilize the error dynamics at the origin. The approach combines three desirable features, that is, the theoretical foundation of the method, the adoption of a scalar synchronizing signal, and the exact analytical solution of the fractional error system written in terms of Mittag-Leffler function. Finally, the projective synchronization of the fractional-order hyperchaotic R?ssler systems is illustrated in detail.     

Complete (anti-)synchronization of chaotic systems with fully uncertain parameters by adaptive control

This paper addresses a unified mathematical expression describing a class of chaotic systems, for which the problem of synchronization and anti-synchronization between different chaotic systems with fully uncertain parameters and different structure are studied. Based on the Lyapunov stability theory, a novel, simple, and systemic adaptive synchronization controller is designated, the analytic expression of the controller and the adaptive laws of parameters are developed. Moreover, the proposed scheme can be extended to anti-synchronize a class of chaotic systems. Two chaotic systems with different structure and fully uncertain parameters are employed as the examples to show the effectiveness of the proposed adaptive synchronization and anti-synchronization schemes. Additionally, the robustness and noise immunity of the adaptive synchronization scheme is investigated by measuring the mean squared error of the systems.     

Application of Takagi–Sugeno fuzzy model to a class of chaotic synchronization and anti-synchronization

In this study, we investigate a class of chaotic synchronization and anti-synchronization with stochastic parameters. A controller is composed of a compensation controller and a fuzzy controller which is designed based on fractional stability theory. Three typical examples, including the synchronization between an integer-order Chen system and a fractional-order Lü system, the anti-synchronization of different 4D fractional-order hyperchaotic systems with non-identical orders, and the synchronization between a 3D integer-order chaotic system and a 4D fractional-order hyperchaos system, are presented to illustrate the effectiveness of the controller. The numerical simulation results and theoretical analysis both demonstrate the effectiveness of the proposed approach. Overall, this study presents new insights concerning the concepts of synchronization and anti-synchronization, synchronization and control, the relationship of fractional and integer order nonlinear systems.     

Adaptive bidirectionally coupled synchronization of?chaotic?systems with unknown parameters

This paper investigates the chaos synchronization of two bidirectionally coupled chaotic systems. In comparison with previous methods (identical bidirectionally coupled synchronization), the present control scheme is different bidirectionally coupled synchronization, which includes different complete bidirectionally coupled synchronization and different partial bidirectionally coupled synchronization. Based on the Lasalle invariance principle, adaptive schemes are designed to make two different bidirectionally coupled chaotic systems asymptotically synchronized, and unknown parameters are identified simultaneously in the process of synchronization. Theoretical analysis and numerical simulations are shown to verify the results.     

Intelligent quadratic optimal synchronization of?uncertain chaotic systems via LMI approach

This paper proposes an intelligent quadratic optimal control scheme via linear matrix inequality (LMI) approach for the synchronization of uncertain chaotic systems with both external disturbances and parametric perturbations. First, a four-layered neural fuzzy network (NFN) identifier is constructed to estimate system nonlinear dynamics. Based on the NFN identifier, an intelligent quadratic optimal controller is developed with robust hybrid control scheme, in which H _?? optimal control and variable structure control (VSC) are embedded to attenuate the effects of external disturbances and parametric perturbations. The adaptive tuning laws of network parameters are derived in the sense of the Lyapunov synthesis approach to ensure network convergence, and the sufficient criterion for existence of the controller is formulated in the linear matrix inequality (LMI) form to guarantee the quadratic optimal synchronization performance. Finally, a numerical simulation example is illustrated by the chaotic Chua??s circuit system to demonstrate the effectiveness of our scheme.     

Chaos synchronization of two different chaotic complex Chen and Lü systems

This paper investigates the phenomenon of chaos synchronization of two different chaotic complex systems of the Chen and Lü type via the methods of active control and global synchronization. In this regard, it generalizes earlier work on the synchronization of two identical oscillators in cases where the drive and response systems are different, the parameter space is larger, and the dimensionality increases due to the complexification of the dependent variables. The idea of chaos synchronization is to use the output of the drive system to control the response system so that the output of the response system converges to the output of the drive system as time increases. Lyapunov functions are derived to prove that the differences in the dynamics of the two systems converge to zero exponentially fast, explicit expressions are given for the control functions and numerical simulations are presented to illustrate the success of our chaos synchronization techniques. We also point out that the global synchronization method is better suited for synchronizing identical chaotic oscillators, as it has serious limitations when applied to the case where the drive and response systems are different.     

Multi-switching synchronization of chaotic system with adaptive controllers and unknown parameters

Using adaptive control techniques, we investigate the multi-switching synchronization of chaotic systems with parameters unknown. Based upon the Lyapunov stability theory, we design the controllers and updating laws of different switching, and it is extended to investigate the synchronization problems with different combinations of slave states with master systems. We take the Lorenz system and the Chen system as an example to analyze the multi-switching synchronization process of different structures of chaotic systems. Finally, numerical simulations have shown the effectiveness of the method.     

Finite-time synchronization control for a class of perturbed nonlinear systems with fixed convergence time and hysteresis quantizer: applied to Genesio–Tesi chaotic system

In the present article, a terminal sliding mode control strategy has been proposed in order to address the synchronization problem for a class of perturbed nonlinear systems with fixed convergence time and input quantization. The proposed protocol guarantees the fixed-time convergence of the sliding manifold to the origin, which means that the convergence time of the proposed sliding manifold does not change on the variations of initial values, different from typical control methods. Here, the hysteresis quantizer, as a specific type of quantizer with nonlinear sector-bounded, is applied in order to quantize the input signal. The proposed quantized control scheme vigorously prevents the potential adverse chattering phenomenon which is experienced in the common quantization methods. The proposed controller does not need the limiting criteria related to considered parameters of quantization compared to recent control approaches. Finally, the designed controller is implemented on the perturbed Genesio–Tesi (G–T) chaotic systems to verify effectiveness and strength of the proposed method.