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.     

Linearly coupled synchronization of the unified chaotic systems and the Lorenz systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems and two linearly coupled Lorenz systems. Some sufficient conditions for synchronization are attained through rigorous mathematical theory. Compared with the results in the reference [Chaos, Solitons & Fractals 2002;14:529], the sufficient condition for the synchronization of two linearly coupled Lorenz systems is simpler and less conservative. Numerical simulations are provided for illustration and verification.     

Designing synchronization schemes for chaotic fractional-order unified systems

Synchronization in chaotic fractional-order differential systems is studied both theoretically and numerically. Two schemes are designed to achieve chaos synchronization of so-called unified chaotic systems and the corresponding numerical algorithms are established. Some sufficient conditions on synchronization are also derived based on the Laplace transformation theory. Computer simulations are used for demonstration.     

Generalized synchronization in linearly coupled time periodic systems

We consider the synchronization of a network of linearly coupled and not necessarily identical oscillators. We present an approach to the existence of the synchronization manifold which is based on some results developed by R. Smith for the study of periodic solutions of ODEs. Our framework allows the study of a large class of systems and does not assume that they are small perturbations of linear systems. Moreover, it provides a practical way to compute estimations on the parameters of the system for which generalized synchronization occurs. Additionally, we give a new proof of the main result of R. Smith on invariant manifolds using Wazewski's principle. Several examples of application are presented.     

Fast synchronization of directionally coupled chaotic systems

This paper studies the fast synchronization of directionally coupled chaotic systems under a chained interaction topology. Firstly, by applying finite-time stability theory, it is shown that all chaotic systems can achieve synchronization in finite time as long as the coupling strength is strong enough. Secondly, it is proved that the settling times are determined by the interaction strength, system parameters and initial conditions of the chaotic systems. Furthermore, it is found that the settling times are mainly dependent on the bounded value and dimension of the coupled chaotic systems when the individual chaotic sub-system is bounded. Finally, illustrative examples and numerical simulations are given to show the correctness of theoretical results.     

Finite-time synchronization of uncertain unified chaotic systems based on CLF

This paper studies the problem of finite-time synchronization for the unified chaotic systems. We prove that global finite-time synchronization can be achieved for unified chaotic systems which have uncertain parameters. Simulation results for Lorenz, Lü and Chen chaotic systems are provided to illustrate the effectiveness of the proposed scheme.     

Some simple global synchronization criterions for coupled time-varying chaotic systems

Based on the Lyapunov stabilization theory and matrix measure, this paper proposes some simple generic criterions of global chaos synchronization between two coupled time-varying chaotic systems from a unidirectional linear error feedback coupling approach. These simple criterions are applicable to some typical chaotic systems with different types of nonlinearity, such as the original Chua’s circuit and the Rössler chaotic system. The coupling parameters are determined according to the new criterion so as to ensure the coupled systems’ global chaos synchronization.     

Global impulsive exponential synchronization of time-delayed coupled chaotic systems

In this paper, the impulsive exponential synchronization problem for time-delayed coupled chaotic systems is investigated. By establishing an impulsive differential delay inequality and using the property of P-cone, some simple conditions of impulsive exponential synchronization of two coupled chaotic systems are derived. To illustrate the effectiveness of the new scheme, some numerical examples are given.     

Stability preserving model reduction for linearly coupled linear time-invariant systems

We develop a stability preserving model reduction method for linearly coupled linear time-invariant (LTI) systems. The method extends the work of Monshizadeh et al. for multi-agent systems with identical LTI agents. They propose using Bounded Real Balanced Truncation to preserve a sufficient condition for stability of the coupled system. Here, we extend this idea to arbitrary linearly coupled LTI systems using the sufficient condition for stability introduced by Reis and Stykel. The model reduction error bounds for this method also follow from results of Reis and Stykel, which allows the adaptive choice of reduced orders. We demonstrate the method on Reis's and Stykel's coupled string-beam example. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A simple criterion for impulsive chaotic synchronization

In the paper, impulsive synchronization of the two identical chaotic systems with different initial conditions is studied. The sufficient conditions on feedback strength and impulsive interval are established to guarantee the synchronization. The method is simple and the relevant mathematical proof is presented. Numerical simulations show the effectiveness of the method.     

A unified approach for impulsive lag synchronization of chaotic systems with time delay

In this paper, we propose a unified approach for impulsive lag-synchronization of a class of chaotic systems with time delay by employing the stability theory of impulsive delayed differential equations. Three well-known delayed chaotic systems are presented to illustrate our results. Also, the estimates of the stable regions for these systems are given, respectively.     

Global synchronization criterion and adaptive synchronization for new chaotic system

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.     

Global synchronization of three coupled chaotic systems with ring connection

This paper studies chaos synchronization of three coupled chaos systems with ring connection. New generic criteria of global chaos synchronization are proposed respectively according to the way of coupling (unidirectional or bidirectional). As an example, The criteria are successfully applied to three coupled identical Lorenz systems. Numerical simulation are shown for demonstration.     

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.     

Adaptive synchronization for a unified chaotic system with uncertainty

A new adaptive controller is designed to synchronize of a unified chaotic system with uncertainty (unknown parameter, noise perturbation, etc.). It is implemented by using variable structure control. The controller designed here only uses the derivative information of the uncertainty. Even if the uncertainty is time-varying or unbounded, as long as its derivative is bounded, the adaptive controller can guarantee the synchronization of the unified chaotic system with uncertainty. Finally, digital simulation is carried out for Lorenz system, and the results verify the effectiveness of the proposed method.     

Exponential generalized synchronization of uncertain coupled chaotic systems by adaptive control

In this paper, the exponential generalized synchronization for a class of coupled systems with uncertainties is defined. A novel and powerful method is proposed to investigate the generalized synchronization based on the adaptive control technique. According to the Lyapunov stability theory, rigorous proof is given for the exponential stability of error system. In comparison with previous schemes, the presented method shortens the synchronization time and is more applicable in practice. Besides, it is shown that the synchronization effect is robust against the uncertain factors. Some typical chaotic and hyper-chaotic systems are taken as examples to illustrate above approach. The corresponding numerical simulations are demonstrated to verify the effectiveness of proposed method.     

A note on phase synchronization in coupled chaotic fractional order systems

The dynamic behaviors of fractional order systems have received increasing attention in recent years. This paper addresses the reliable phase synchronization problem between two coupled chaotic fractional order systems. An active nonlinear feedback control scheme is constructed to achieve phase synchronization between two coupled chaotic fractional order systems. We investigated the necessary conditions for fractional order Lorenz, Lü and Rössler systems to exhibit chaotic attractor similar to their integer order counterpart. Then, based on the stability results of fractional order systems, sufficient conditions for phase synchronization of the fractional models of Lorenz, Lü and Rössler systems are derived. The synchronization scheme that is simple and global enables synchronization of fractional order chaotic systems to be achieved without the computation of the conditional Lyapunov exponents. Numerical simulations are performed to assess the performance of the presented analysis.     

Some impulsive synchronization criterions for coupled chaotic systems via unidirectional linear error feedback approach

Based on stability theory of impulsive differential equation and new comparison theory of impulsive differential system, we study the chaos impulsive synchronization of two coupled chaotic systems using the unidirectional linear error feedback scheme. Some generic conditions of chaos impulsive synchronization of two coupled chaotic systems are derived, and to apply the conditions to typical chaotic system––the original Chua’s circuit. The example illustrates the effectiveness of the proposed result.     

Function projective synchronization for fractional-order chaotic systems

This letter investigates the function projective synchronization between fractional-order chaotic systems. Based on the stability theory of fractional-order systems and tracking control, a controller for the synchronization of two fractional-order chaotic systems is designed. This technique is applied to achieve synchronization between the fractional-order Lorenz systems with different orders, and achieve synchronization between the fractional-order Lorenz system and fractional-order Chen system. The numerical simulations demonstrate the validity and feasibility of the proposed method.     

Partial synchronization in coupled chemical chaotic oscillators

In this paper we investigate the problem of partial synchronization in diffusively coupled chemical chaotic oscillators with zero-flux boundary conditions. The dynamical properties of the chemical system which oscillates with Uniform Phase evolution, yet has Chaotic Amplitudes (UPCA) are first discussed. By combining numerical and analytical methods, the impossibility of full global synchronization in a network of two or three coupled chemical oscillators is discovered. Mathematically, stable partial synchronization corresponds to convergence to a linear invariant manifold of the global state space. The sufficient conditions for exponential stability of the invariant manifold in a network of three coupled chemical oscillators are obtained via the nonlinear contraction principle.