Adaptive control and synchronization of identical new chaotic flows with unknown parameters via single input

The single input linear feedback control for synchronizing two identical new 3D chaotic flows reported by Li et al. [X.F. Li, K.E. Chlouverakis, D.L. Xu, Nonlinear dynamics and circuit realization of a new chaotic flow: a variant of Lorenz, Chen and Lü, Nonlinear Analysis RWA 10 (4) (2009) 2357-2368] is proposed in this paper. Sufficient conditions of synchronization are obtained for both linear feedback and adaptive control approaches. The problem of adaptive synchronization between two nearly identical chaotic systems with unknown parameters is also studied. Based on the Lyapunov stability theory, two kinds of single input adaptive synchronization controllers are designed and the adaptive parameter update laws are developed.     

Exponential stability criteria of uncertain systems with multiple time delays

The exponential stability (with convergence rate α) of uncertain linear systems with multiple time delays is studied in this paper. Using the characteristic function of linear time-delay system, stability criteria are derived to guarantee α-stability. Sufficient conditions are also obtained for exponential stability of uncertain parametric systems with multiple time delays. For two-dimensional time-invariant system with multiple time delays, the proposed stability criteria are shown to be less conservative than those in the literature. Numerical examples are given to illustrate the validity of our new stability criteria.     

Modified projective synchronization of uncertain fractional order hyperchaotic systems

Base on the stability theory of fractional order system, this work mainly investigates modified projective synchronization of two fractional order hyperchaotic systems with unknown parameters. A controller is designed for synchronization of two different fractional order hyperchaotic systems. The method is successfully applied to modified projective synchronization between fractional order Rössler hyperchaotic system and fractional order Chen hyperchaotic system, and numerical simulations illustrate the effectiveness of the obtained results.     

Exponential attractor and its fractal dimension for a second order lattice dynamical system

We construct an exponential attractor for a second order lattice dynamical system with nonlinear damping arising from spatial discretization of wave equations in Rk. And we obtain fractal dimension of the exponential attractor and its finite-dimensional approximation.     

Adaptive impulsive synchronization of uncertain delayed chaotic system with full unknown parameters via discrete‐time drive signals

This article investigates the adaptive impulsive synchronization of delayed chaotic system with full unknown parameters. Aiming at this problem, we propose a new adaptive strategy, in which both the adaptive–impulsive controller and the parameters adaptive laws are designed via the discrete‐time signals from the drive system. The corresponding theoretical proof is given to guarantee the effectiveness of the proposed strategy. Moreover, the concrete adaptive strategies are achieved for delayed Hopfield neural network, optical Ikeda system and the well‐known delayed Lü chaotic system. As expected, numerical simulations show the effectiveness of the proposed strategy. This method has potential applications in parameters estimation, secure communication, and cryptanalysis when only discrete signals are transmitted in communication channel. © 2014 Wiley Periodicals, Inc. Complexity 21: 43–51, 2016     

Exponential synchronization of stochastic perturbed complex networks with time-varying delays via periodically intermittent pinning

In this paper, the exponential synchronization is investigated for stochastic complex networks with time-varying delays via periodically intermittent pinning control. By utilizing the Lyapunov stability theory, stochastic analysis theory as well as linear matrix inequalities (LMI), the sufficient conditions are derived to guarantee the exponential synchronization. Furthermore, the complex networks considered in this paper are more general than the models in previous works. Therefore, the application scope is enlarged. And the result is computationally efficient for the obtained condition. The numerical simulation is provided to show the effectiveness of the theoretical results.     

Exponential synchronization of stochastic coupled oscillators networks with delays

This paper investigates the exponential synchronization problem of coupled oscillators networks with disturbances and time-varying delays. On basis of graph theory and stochastic analysis theory, a feedback control law is designed to achieve exponential synchronization. By constructing a global Lyapunov function for error network, both pth moment exponential synchronization and almost sure exponential synchronization of drive-response networks are obtained. Finally, a numerical example is given to show the effectiveness of the proposed criteria.     

Circuit realization,chaos synchronization and estimation of parameters of a hyperchaotic system with unknown parameters

In this article, the adaptive chaos synchronization technique is implemented by an electronic circuit and applied to the hyperchaotic system proposed by Chen et al. We consider the more realistic and practical case where all the parameters of the master system are unknowns. We propose and implement an electronic circuit that performs the estimation of the unknown parameters and the updating of the parameters of the slave system automatically, and hence it achieves the synchronization. To the best of our knowledge, this is the first attempt to implement a circuit that estimates the values of the unknown parameters of chaotic system and achieves synchronization. The proposed circuit has a variety of suitable real applications related to chaos encryption and cryptography. The outputs of the implemented circuits and numerical simulation results are shown to view the performance of the synchronized system and the proposed circuit.     

Adaptive synchronization of two novel different hyperchaotic systems with partly uncertain parameters

This paper investigates adaptive synchronization between two novel different hyperchaotic systems with partly uncertain parameters. Based on the Lyapunov stability theorem and the adaptive control theory, synchronization between these two hyperchaotic systems is achieved by proposing a new adaptive controller and a parameter estimation update law. Numerical simulations are presented to demonstrate the analytical results.     

Projective synchronization between different fractional‐order hyperchaotic systems with uncertain parameters using proposed modified adaptive projective synchronization technique

In the present article, the authors have proposed a modified projective adaptive synchronization technique for fractional‐order chaotic systems. The adaptive projective synchronization controller and identification parameters law are developed on the basis of Lyapunov direct stability theory. The proposed method is successfully applied for the projective synchronization between fractional‐order hyperchaotic Lü system as drive system and fractional‐order hyperchaotic Lorenz chaotic system as response system. A comparison between the effects on synchronization time due to the presence of fractional‐order time derivatives for modified projective synchronization method and proposed modified adaptive projective synchronization technique is the key feature of the present article. Numerical simulation results, which are carried out using Adams–Boshforth–Moulton method show that the proposed technique is effective, convenient and also faster for projective synchronization of fractional‐order nonlinear dynamical systems. Copyright © 2013 John Wiley & Sons, Ltd.     

Exponential synchronization of discontinuous chaotic systems via delayed impulsive control and its application to secure communication

This paper investigates drive-response synchronization of chaotic systems with discontinuous right-hand side. Firstly, a general model is proposed to describe most of known discontinuous chaotic system with or without time-varying delay. An uniform impulsive controller with multiple unknown time-varying delays is designed such that the response system can be globally exponentially synchronized with the drive system. By utilizing a new lemma on impulsive differential inequality and the Lyapunov functional method, several synchronization criteria are obtained through rigorous mathematical proofs. Results of this paper are universal and can be applied to continuous chaotic systems. Moreover, numerical examples including discontinuous chaotic Chen system, memristor-based Chua’s circuit, and neural networks with discontinuous activations are given to verify the effectiveness of the theoretical results. Application of the obtained results to secure communication is also demonstrated in this paper.     

Exponential synchronization of Markovian jumping chaotic neural networks with sampled-data and saturating actuators

This paper deals with the problem of exponential synchronization of Markovian jumping chaotic neural networks with saturating actuators using a sampled-data controller. By constructing a proper Lyapunov–Krasovskii functional (LKF) with triple integral terms, and employing Jensen’s inequality, some new sufficient conditions for the exponential synchronization of considered chaotic neural networks are derived in terms of linear matrix inequalities (LMIs). The obtained LMIs can be easily solved by any of the available software. Finally, the numerical examples are provided to demonstrate the effectiveness of our theoretical results.     

Exponential synchronization of fuzzy cellular neural networks with mixed delays and general boundary conditions

In this paper, we study exponential synchronization of delayed reaction-diffusion fuzzy cellular neural networks with general boundary conditions. By using Sobolev inequality techniques and constructing suitable Lyapunov functional, some sufficient conditions are given to ensure the exponential synchronization of the drive-response delayed fuzzy cellular neural networks with general boundary conditions. Finally, an example is given to verify the theoretical analysis.     

Function vector synchronization based on fuzzy control for uncertain chaotic systems with dead‐zone nonlinearities

In this article, a fuzzy adaptive control scheme is designed to achieve a function vector synchronization behavior between two identical or different chaotic (or hyperchaotic) systems in the presence of unknown dynamic disturbances and input nonlinearities (dead‐zone and sector nonlinearities). This proposed synchronization scheme can be considered as a generalization of many existing projective synchronization schemes (namely the function projective synchronization, the modified projective synchronization, generalized projective synchronization, and so forth) in the sense that the master and slave outputs are assumed to be some general function vectors. To practically deal with the input nonlinearities, the adaptive fuzzy control system is designed in a variable‐structure framework. The fuzzy systems are used to appropriately approximate the uncertain nonlinear functions. A Lyapunov approach is used to prove the boundedness of all signals of the closed‐loop control system as well as the exponential convergence of the corresponding synchronization errors to an adjustable region. The synchronization between two identical systems (chaotic satellite systems) and two different systems (chaotic Chen and Lü systems) are taken as two illustrative examples to show the effectiveness of the proposed method. © 2015 Wiley Periodicals, Inc. Complexity 21: 234–249, 2016     

Existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays

By using the continuation theorem due to Mawhin and Gaines, some analysis techniques and the Lyapunov functional method, the sufficient conditions ensuring the existence of an exponential periodic attractor of a class of impulsive differential equations with time-varying delays are established. The results are interesting and very different from previously known results [Xia and Wong, 2009 [7]; Tan and Tan, 2009 [19]; Huang et al., 2005 [20]; Liu and Huang, 2006 [22]]. Finally, applications and an example are given to illustrate the effectiveness of the results.     

A general scheme for Q-S synchronization of chaotic systems with unknown parameters and scaling functions

This work investigates Q-S synchronization of non-identical chaotic systems with unknown parameters and scaling function. The sufficient conditions for achieving Q-S synchronization with a double-desired scaling function of two different chaotic systems (including different dimensional systems) are derived based on the Lyapunov stability theory. By the adaptive control technique, the corresponding parameter update laws are proposed such that the Q-S synchronization of non-identical chaotic systems is to be obtained. Two illustrative numerical simulations are also given to demonstrate the effectiveness of the proposed scheme.     

Exponential stabilization of stochastic interval system with time dependent parameters

There has been growing interest in analyzing stability in design controls of stochastic systems. This interest arises out of the need to develop robust control strategies for systems with uncertain dynamics. This paper is concerned with the examination of conditions under which the desired structure of a stochastic interval system with time dependent parameters is stabilizable. Necessary and sufficient condition under which two-level preconditioner guarantees quadratic mean exponential stability of the desired structure of uncontrolled stochastic interval system is presented. Sufficient condition for exponential stability of the equilibrium solution of uncontrolled stochastic interval system is also presented.     

Chaotic bursting lag synchronization of Hindmarsh–Rose system via a single controller

Chaotic bursting lag synchronization of Hindmarsh–Rose system is investigated. Two lag synchronization schemes with only a single controller are proposed to synchronize Hindmarsh–Rose chaotic system via back stepping method. Especially in the second scheme, only one state variable is contained in the controller. Based on Lyapunov stability theory, the sufficient conditions for synchronization are obtained analytically in both cases. Finally, numerical simulations are provided to show the effectiveness of the developed methods.     

Adaptive control and synchronization of a new modified hyperchaotic Lü system with uncertain parameters

This paper addresses problems of control and synchronization for a new modified hyperchaotic Lü system with uncertain parameters. This new modified uncertain hyperchaotic Lü system is stabilized to its unique unstable equilibrium by using adaptive control. Furthermore, an adaptive control law and a parameter estimation update law are derived to synchronize two identical modified hyperchaotic Lü systems with uncertain parameters. Numerical examples are proposed to demonstrate and verify the theoretical analysis.     

Uniform attractor for nonautonomous incompressible non-Newtonian fluid with a new class of external forces

This paper is joint with [27]. The authors prove in this article the existence and reveal its structure of uniform attractor for a two-dimensional nonautonomous incompressible non-Newtonian fluid with a new class of external forces.