Stochastic linear generalized synchronization of chaotic systems via robust control

In this Letter, based on robust control, we provide a general theoretical result on stochastic linear generalized synchronization (GS) of chaotic systems. Given a driving system with noise perturbations and a linear synchronization function, a response system is developed easily according to the scheme derived here. By introducing the Lyapunov stability theory and linear matrix inequalities (LMIs), the condition for synchronization is proved to be effective. Finally, the Lorenz system is taken for illustration and verification.     

Projective synchronization of a class of delayed chaotic systems via impulsive control

The Letter studies the projective synchronization of a class of delayed chaotic systems. The drive-response system can be synchronized to within a desired scaling factor via impulsive control. Some sufficient conditions are derived by the stability analysis of the impulsive functional differential equations. An illustrative example is provided to show the effectiveness and feasibility of the proposed method and results.     

Stability criterion for synchronization of chaotic systems using linear feedback control

This Letter introduces linear balanced feedback control scheme to design controller for the synchronization of two identical chaotic systems based on Lyapunov stability theory and constrained extreme approach. The technique is applied to synchronizing two identical four-scroll chaotic systems and guiding balanced feedback gains design. In accordance with the result of the analysis, an adaptive control scheme is proposed for chaos synchronization when the parametric variations of the response system are uncertain. The feasibility and effectiveness of the proposed synchronization scheme are verified via numerical simulations.     

Adaptive full state hybrid projective synchronization of chaotic systems with the same and different order

This Letter further investigates the full state hybrid projective synchronization (FSHPS) of chaotic and hyper-chaotic systems with fully unknown parameters. Based on the Lyapunov stability theory, a unified adaptive controller and parameters update law can be designed for achieving the FSHPS of chaotic and/or hyper-chaotic systems with the same and different order. Especially, for two chaotic systems with different order, reduced order MFSHPS (an acronym for modified full state hybrid projective synchronization) and increased order MFSHPS are first studied in this Letter. Five groups numerical simulations are provided to verify the effectiveness of the proposed scheme. In addition, the proposed FSHPS scheme is quite robust against the effect of noise.     

Impulsive control of projective synchronization in chaotic systems

Scaling factor of projective synchronization in coupled partially linear chaotic systems is hardly predictable. To control projective synchronization of chaotic systems in a preferred way, an impulsive control scheme is introduced to direct the scaling factor onto a desired value. The control approach is derived from the impulsive differential equation theory. Numerical simulations on the chaotic Lorenz system are illustrated to verify the theoretical results. Furthermore, some interesting and surprising numerical results are discussed.     

Anticipating synchronization of chaotic Lur'e systems

In this paper we consider the anticipating synchronization of chaotic time-delayed Lur'e-type systems in a master-slave setting. We introduce three scenarios for anticipating synchronization, and give sufficient conditions for the existence of anticipating synchronizing slave systems in terms of linear matrix inequalities. The results obtained are illustrated on a time-delayed Rossler system and a time-delayed Chua oscillator.     

An extended active control for chaos synchronization

By introducing a control strength matrix, the active control theory in chaotic synchronization is developed. With this extended method, chaos complete synchronization can be achieved more easily, i.e., a much smaller control signal is enough to reach synchronization in most cases. Numerical simulations on Rossler, Liu's four-scroll, and Chen system confirmed this and show that the synchronization result depends on the control strength significantly. Especially, in the case of Liu and Chen systems, the response systems' largest Lyapunov exponents' variation with the control strength is not monotone and there exist minima. It is novel for Chen system that the synchronization speed with a special small strength is higher than that of the usual active control which, as a special case of the extended method, has a much larger control strength. All these results indicate that the control strength is an important factor in the actual synchronization. So, with this extended active control, one can make a better and more practical synchronization scheme by adjusting the control strength matrix.     

Fuzzy adaptive synchronization of uncertain chaotic systems via delayed feedback control

Based on the T-S fuzzy model and the delayed feedback control (DFC) scheme, this Letter presents a robust synchronization strategy for a class of chaotic system with unknown parameters and disturbances. Being the response system, the designed robust observer can adaptively track the drive system globally. The T-S fuzzy model of the 4D chaotic system (Lorenz-Stenflo) is developed as an example for illustration. Numerical simulations are shown to verify the results.     

Estimation of chaotic coupled map lattices using symbolic vector dynamics

In [K. Wang, W.J. Pei, Z.Y. He, Y.M. Cheung, Phys. Lett. A 367 (2007) 316], an original symbolic vector dynamics based method has been proposed for initial condition estimation in additive white Gaussian noisy environment. The estimation precision of this estimation method is determined by symbolic errors of the symbolic vector sequence gotten by symbolizing the received signal. This Letter further develops the symbolic vector dynamical estimation method. We correct symbolic errors with backward vector and the estimated values by using different symbols, and thus the estimation precision can be improved. Both theoretical and experimental results show that this algorithm enables us to recover initial condition of coupled map lattice exactly in both noisy and noise free cases. Therefore, we provide novel analytical techniques for understanding turbulences in coupled map lattice.     

On the synchronization of a hyperchaotic system based on adaptive method

This Letter researches the adaptive synchronization of a hyperchaotic system. Based on Lyapunov method, a general adaptive method via a driving vector and an original adaptive method via a single driving variable are presented to achieve the chaos synchronization. Especially, the latter makes the structure of adaptive controller simple and has more practical value compared with the former. Some typical numerical examples are included to demonstrate the effectiveness of the method proposed.     

Impulsive synchronization of chaotic systems

The issue of impulsive synchronization of a class of chaotic systems is investigated. Based on the impulsive theory and linear matrix inequality technique, some less conservative and easily verified criteria for impulsive synchronization of chaotic systems are derived. The proposed method is applied to the original Chua oscillators, and the corresponding synchronization conditions are obtained. Moreover, the boundary of the stable region is also estimated in terms of the equidistant impulse interval. The effectiveness of our method is shown by computer simulation.     

Master-slave synchronization of Lur'e systems with sector and slope restricted nonlinearities

This Letter presents a synchronization method for Lur'e systems with sector and slope restricted nonlinearities. A static error feedback controller based on the Lyapunov stability theory is proposed for asymptotic synchronization. The Lyapunov function candidate is chosen as a quadratic form of the error states and nonlinear functions of the systems. The nonlinearities are expressed as convex combinations of sector and slope bounds by using convex properties of the nonlinear function so that equality constraints are converted into inequality constraints. Then, the feedback gain matrix is derived through a linear matrix inequality (LMI) formulation. Finally, a numerical example shows the effectiveness of the proposed method.     

Exponential synchronization of chaotic Lur’e systems with time-varying delay via sampled-data control

In this paper, we study the exponential synchronization of chaotic Lur＇e systems with time-varying delays via sampled-data control by using sector nonlinearties. In order to make full use of information about sampling intervals and interval time-varying delays, new Lyapunov-Krasovskii functionals with triple integral terms are introduced. Based on the convex combination technique, two kinds of synchronization criteria are derived in terms of linear matrix inequal- ities, which can be efficiently solved via standard numerical software. Finally, three numerical examples are provided to demonstrate the less conservatism and effectiveness of the proposed results.     

Structure identification and adaptive synchronization of uncertain general complex dynamical networks

This Letter proposes an approach to identify the topological structure and unknown parameters for uncertain general complex networks simultaneously. By designing effective adaptive controllers, we achieve synchronization between two complex networks. The unknown network topological structure and system parameters of uncertain general complex dynamical networks are identified simultaneously in the process of synchronization. Several useful criteria for synchronization are given. Finally, an illustrative example is presented to demonstrate the application of the theoretical results.     

Observer-based synchronization in complex dynamical networks with nonsymmetric coupling

Based on a general complex dynamical network model with nonsymmetric coupling, some criteria for synchronization are proposed based on the approach of state observer design. Unlike the nonobserver-based dynamical networks, where the coupling between two connected nodes is defined by an inner coupling matrix and full state coupling is typically needed, in this paper, smaller amount of coupling variables or even only a scalar output signal of each node is needed to synchronize the network. Unlike the commonly researched complex network model, where the coupling between nodes is symmetric, here, in our network model, the coupling configuration matrix is not assumed to be symmetric and may have complex eigenvalues. The matrix Jordan canonical formalization method is used instead of the matrix diagonalization method, so in our synchronization criteria, the coupling configuration matrix is not required to be diagonalizable. Especially, the proposed step-by-step approach is simpler in computation than the existent ones, which usually rely heavily on numerical toolbox, and may be done by hand completely. An example is given to illustrate the step-by-step approach, in which each node is a two-dimensional dynamical limit cycle oscillator system consisting of a two-cell cellular neural network, and numerical simulations are also done to verify the results of design.     

Adaptive synchronization between two complex networks with nonidentical topological structures

This paper addresses the theoretical analysis of synchronization between two complex networks with nonidentical topological structures. By designing effective adaptive controllers, we achieve synchronization between two complex networks. Both the cases of identical and nonidentical network topological structures are considered and several useful criteria for synchronization are given. Illustrative examples are presented to demonstrate the application of the theoretical results.     

Pinning a complex dynamical network via impulsive control

Complex dynamical networks are being studied across many fields of science and engineering today. The issue of controlling a network to the desired state has attracted increasing attention. In this Letter, we investigate the problem of pinning a complex dynamical network to the solution of an uncoupled system. Our strategy is to apply impulsive control to a small fraction of network nodes. Based on the Lyapunov stability theory, we prove that the theoretical results derived here are effective. In addition, a B-A scale-free network with 20 nodes is taken for illustration and verification.     

Function projective synchronization of different chaotic systems with uncertain parameters

This Letter investigates the function projective synchronization of different chaotic systems with unknown parameters. By Lyapunov stability theory, the adaptive control law and the parameter update law are derived to make the states of two different chaotic systems asymptotically synchronized up to a desired scaling function. Numerical simulations on Lorenz system and Newton-Leipnik system are presented to verify the effectiveness of the proposed scheme.     

Projective Synchronization in Drive--Response Networks via Impulsive Control

Impulsive projective synchronization in 1 ＋N coupled chaotic systems are investigated with the drive-response dynamical network （DRDN） model. Based on impulsive stability theory, some simple but less conservative criteria axe achieved for projective synchronization in DRDNs. Furthermore, impulsive pinning scheme is also adopted to direct the scaring factor onto the desired value. Numerical simulations on generalized chaotic unified system axe illustrated to verify the theoretical results.     

The impulsive control synchronization of the drive-response complex system

This Letter investigates projective synchronization between the drive system and response complex dynamical system. An impulsive control scheme is adapted to synchronize the drive-response dynamical system to a desired scalar factor. By using the stability theory of the impulsive differential equation, the criteria for the projective synchronization are derived. The feasibility of the impulsive control of the projective synchronization is demonstrated in the drive-response dynamical system.