Exponential synchronization of chaotic systems subject to uncertainties in the control input

A sliding mode control technique is introduced for exponential synchronization of chaotic systems. These systems are described by a general form including matched and unmatched nonlinear functions. A new hitting-free switching surface of proportional-integral type is proposed. This type of switching surface is without the hitting process if the attraction of sliding manifold is ensured. This property makes it easy to exponentially synchronize the master-slave chaotic systems. Based on this switching surface, a robust sliding mode controller (SMC) is derived to guarantee the attraction of sliding manifold even when the system is subjected to input uncertainties. An example is included to illustrate the results developed in this paper.     

Robust chaos synchronization of four-dimensional energy resource systems subject to unmatched uncertainties

This paper investigates the robust chaos synchronization problem for the four-dimensional energy resource systems. Based on the sliding mode control (SMC) technique, this approach only uses a single controller to achieve chaos synchronization, which reduces the cost and complexity for synchronization control implementation. As expected, the error states can be driven to zero or into predictable bounds for matched and unmatched perturbations, respectively. Numerical simulation results, which fully coincide with theoretical results, are presented to demonstrate the obtained results.     

Robust adaptive synchronization of different uncertain chaotic systems subject to input nonlinearity

In this paper, an adaptive controller is designed to ensure robust synchronization of two different chaotic systems with input nonlinearities. For this purpose, a stable sliding surface is defined and an adaptive sliding mode controller is designed to achieve robust synchronization of the systems when the control input is influenced through nonlinearities produced by actuator or external uncertainty recourses. The adaptation law guarantees the synchronization assuming of unknown model uncertainty. Furthermore by adding an integrator and incorporating a saturation function in the control law, the chattering phenomenon caused by the sign function is avoided. The simulation results for synchronization of Chua’s circuit and Genesio systems show the efficiency of the proposed technique.     

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.     

Robust control of chaos in Lorenz systems subject to mismatch uncertainties

This paper is concerned with robust control for a class of Lorenz systems subject to mismatch uncertainties. It is implemented by using variable structure control. The proposed variable structure controller ensures the occurrence of the sliding mode for the error dynamics. It is guaranteed that under the proposed control law, uncertain Lorenz systems can drive the system state exactly to some specific points or in a predictable neighborhood of arbitrary desired points in the state space even with mismatch uncertainties, which is not addressed in the literature.     

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.     

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.     

Robust global exponential synchronization of general Lur’e chaotic systems subject to impulsive disturbances and time delays

In this paper, we aim to study the robust global exponential synchronization problem for a general class of Lur’e chaotic systems subject to time delays and impulsive disturbances. Furthermore, we also provide an estimation of the maximum Lyapunov exponent. By using the Lyapunov function method and linear matrix inequality (LMI) technique, sufficient conditions for the robust global exponential synchronization and estimation of its maximum Lyapunov exponent are obtained for the class of Lur’e chaotic systems with and without time delays, respectively. Furthermore, by applying the M-matrix theory, some of these sufficient conditions are shown to be expressible in forms of fairly simple algebraic conditions. For illustration, several examples are solved by using the sufficient conditions obtained.     

Robust synchronization of chaotic non-autonomous systems using adaptive-feedback control

In this paper, we apply the simple adaptive-feedback control scheme to synchronize a class of chaotic non-autonomous systems. Based on the invariance principle of differential equations, some generic sufficient conditions for global asymptotic synchronization are obtained. Unlike the usual linear feedback, the variable feedback strength is automatically adapted to completely synchronize two identical systems and simple to implement in practice. As illustrative examples, synchronization of two parametrically excited chaotic pendulums and that of two 4D new systems are considered here. Numerical simulations show the proposed method is effective and robust against the effect of noise.     

Robust synchronization of fractional-order complex dynamical networks with parametric uncertainties

The study investigates robust synchronization of fractional-order complex dynamical networks with parametric uncertainties. Based on the properties of the kronecker product and the stability of the fractional-order system, the robust synchronization criteria are derived by applying the nonlinear control. These criteria are in the form of linear matrix inequalities which can be readily solved by applying the LMI toolbox. The coupling configuration matrix is not necessary to be symmetric or irreducible, and the inner coupling matrix needs not to be symmetric, diagonal or positive definite. Two numerical examples are provided to demonstrate the validity of the presented synchronization scheme.     

Synchronizing chaotic dynamics with uncertainties using a predictable synchronization delay design

This paper deals with synchronization and optimization problems of second-order chaotic oscillators by applying a novel control scheme. The approach developed considers incomplete state measurements and no detailed model of the systems to guarantee robust stability. This approach includes an uncertainty estimator and leads to a robust predictable feedback control scheme. The synchronization of the ⁶-Duffing and ⁶-Van der Pol oscillators was used as an illustrative example. A fairly good agreement is obtained between the analytical and numerical results.     

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.     

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.     

Adaptive terminal sliding mode control subject to input nonlinearity for synchronization of chaotic gyros

Under the existence of system uncertainties, external disturbances, and input nonlinearity, complete synchronization and anti-synchronization between two chaotic gyros are achieved by introducing a novel adaptive terminal sliding mode (ATSM) controller. In the literature, by taking account of input nonlinearity, the magnitudes of bounded nonlinear dynamics of synchronous error system were required in the designed sliding mode controller. In this study, the proposed ATSM controller associated with time-varying feedback gains can tackle nonlinear dynamics according to the novel adaptive rules. These feedback gains are not necessary to be determined in advance but updated by the adaptive rules without known the magnitudes of bounded nonlinear dynamics, system uncertainties, and external disturbances. Sufficient conditions to guarantee stable synchronization are given in the sense of the Lyapunov stability theorem, and the numerical simulations are performed to verify the effectiveness of presented schemes.     

Exponential synchronization for fractional‐order chaotic systems with mixed uncertainties

This article focuses on the problem of exponential synchronization for fractional‐order chaotic systems via a nonfragile controller. A criterion for α‐exponential stability of an error system is obtained using the drive‐response synchronization concept together with the Lyapunov stability theory and linear matrix inequalities approach. The uncertainty in system is considered with polytopic form together with structured form. The sufficient conditions are derived for two kinds of structured uncertainty, namely, (1) norm bounded one and (2) linear fractional transformation one. Finally, numerical examples are presented by taking the fractional‐order chaotic Lorenz system and fractional‐order chaotic Newton–Leipnik system to illustrate the applicability of the obtained theory. © 2014 Wiley Periodicals, Inc. Complexity 21: 114–125, 2015     

Stability criterion for synchronization of linearly coupled unified chaotic systems

This paper investigates the synchronization of two linearly coupled unified chaotic systems. A new stability criterion for asymptotic synchronization is attained using the Lyapunov stability theory and linear matrix inequality (LMI) approach. A numerical example is given to illuminate the design procedure and advantage of the result derived.     

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.     

Robust synchronization of a class of chaotic systems with disturbance estimation

This paper investigates the robust synchronization problem for a class of chaotic systems with external disturbances. By using disturbance-observer-based control (DOBC) and LMI approach, the disturbance observers are developed to ensure the boundedness of the disturbance error dynamical. Then, by employing the sliding mode control technique, an adaptive control law is established to eliminate the effect of disturbance error to realize synchronization between the master and slave systems. Finally, 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 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.