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.     

Hybrid robust modified function projective lag synchronization in two different dimensional chaotic systems

In this article, a novel synchronization scheme, modified function projective lag synchronization (MFPLS) in two different dimensional chaotic systems with parameter perturbations, is proposed. In the proposed method, the states of two nonidentical chaotic systems with different orders are asymptotically lag synchronized up to a desired scaling function matrix by means of reduced order and increased order, respectively. Furthermore, based on the reality situation, the parameter perturbations are involved, which are assumed to appear in both drive and response systems. With the Lyapunov stability theory, an adaptive controller is designed to achieve MFPLS. Theoretical proof and numerical simulations demonstrate the effectiveness and feasibility of the proposed scheme.     

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.     

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.     

Lag synchronization for fuzzy chaotic system based on fuzzy observer

A new fuzzy observer for lag synchronization is given in this paper. By investi- gating synchronization of chaotic systems, the structure of drive-response lag synchronization for fuzzy chaos system based on fuzzy observer is proposed. A new lag synchronization criterion is derived using the Lyapunov stability theorem, in which control gains are obtained under the LMI condition. The proposed approach is applied to the well-known Chen＇s systems. A simulation example is presented to illustrate its effectiveness.     

Mean square function synchronization of chaotic systems with stochastic effects

In this paper, we give the definition of mean square function synchronization. Secondly, we investigate mean square function synchronization of chaotic systems with stochastic perturbation and unknown parameters. Based on the Lyapunov stability theory, inequality techniques, and the properties of the Weiner process, the controller, and adaptive laws are designed to ensure achieving stochastic synchronization of chaotic systems. A sufficient synchronization condition is given to ensure the chaotic systems to be mean-square stable. Furthermore, a numerical simulation is also given to demonstrate the effectiveness of the proposed scheme.     

Projective synchronization between two different time-delayed chaotic systems using active control approach

In this paper, we investigate the projective synchronization between two different time-delayed chaotic systems. A suitable controller is chosen using the active control approach. We relax some limitations of previous work, where projective synchronization of different chaotic systems can be achieved only in finite dimensional chaotic systems, so we can achieve projective synchronization of different chaotic systems in infinite dimensional chaotic systems. Based on the Lyapunov stability theory, we suggest a generic method to achieve the projective synchronization between two different time-delayed chaotic systems. The validity of the proposed method is demonstrated and verified by observing the projective synchronization between two well-known time-delayed chaotic systems; the Ikeda system and Mackey–Glass system. Numerical simulations fully support the analytical approach.     

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.     

Combination–combination synchronization among four identical or different chaotic systems

Based on one drive system and one response system synchronization model, a new type of combination–combination synchronization is proposed for four identical or different chaotic systems. According to the Lyapunov stability theorem and adaptive control, numerical simulations for four identical or different chaotic systems with different initial conditions are discussed to show the effectiveness of the proposed method. Synchronization about combination of two drive systems and combination of two response systems is the main contribution of this paper, which can be extended to three or more chaotic systems. A universal combination of drive systems and response systems model and a universal adaptive controller may be designed to our intelligent application by our synchronization design.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain Lorenz system, and the R(o)ssler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

Linear feedback controller design method for time-delay chaotic systems

Based on the stability theory of time-delay systems, this paper discusses chaos control and synchronization problem of time-delay chaotic systems. Through the combining of a new theorem and the characters of the chaotic system, we have designed a linear controller to realize chaos control and synchronization for Lorenz system with time-varying lags. Finally, numerical simulations are provided to verify the effectiveness and feasibility of the developed method.     

Adaptive lag synchronization in coupled chaotic systems with unidirectional delay feedback

Lag synchronization in unidirectionally coupled chaotic systems is investigated in this paper. Based on the invariance principle of differential equations, a new adaptive delay feedback scheme is proposed to realize the lag synchronization effectively in the coupled chaotic systems. As an example, numerical simulations for the coupled Hindmarsh-Rose (HR) neuron models are conducted, which is in good agreement with the theoretical analysis. More interestingly, it is found that there is a fine U-shaped structure in the lag synchronization curve for the HR neuron model. Furthermore, lag synchronization and the corresponding U-shaped structure are robust against the small mismatch of parameters and noisy disturbances.     

Chaos synchronization of Rikitake chaotic attractor using the passive control technique

Chaos synchronization of Rikitake system applying the passive control method is investigated in this paper. Based on the passive technique, the passive controllers are designed. The nonlinear controller for the synchronization of two identical Rikitake systems or two different chaotic systems is simple and convenient to realize. Both theoretical analysis and numerical results show the effectiveness of the proposed method.     

Robust adaptive synchronization for a class of chaotic systems with actuator failures and nonlinear uncertainty

This paper is concerned with the robust adaptive synchronization problem for a class of chaotic systems with actuator failures and unknown nonlinear uncertainty. Combining adaptive method and linear matrix inequality (LMI) technique, a novel type of robust adaptive reliable synchronization controller is proposed, which can eliminate the effect of actuator fault and nonlinear uncertainty on systems. After solving a set of LMIs, synchronization error between the master chaotic and slave chaotic systems can converge asymptotically to zero. Finally, illustrate examples about chaotic Chua’s circuit system and Lorenz systems are provided to demonstrate the effectiveness and applicability of the proposed design method.     

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.     

On fractional difference logistic maps: Dynamic analysis and synchronous control

This paper investigates a logistic map derived from a difference equation in the framework of discrete fractional calculus. Through the Poincaré plots and Julia sets, the map’s chaotic and fractal characteristics are studied comparing with those of a quadratic map to be proposed. The memory effect of fractional difference maps is reflected in these dynamics, and some reasonable explanations are given by combining with quantitative analysis. A coupled controller is designed to realize synchronization between fractional difference logistic map and fractional difference quadratic map.

     

Finite-time combination-combination synchronization of four different chaotic systems with unknown parameters via sliding mode control

In this paper, we apply the nonsingular terminal sliding mode control technique to realize the novel combination-combination synchronization between combination of two chaotic systems as drive system and combination of two chaotic systems as response system with unknown parameters in a finite time. On the basic of the adaptive laws and finite-time stability theory, an adaptive combination sliding mode controller is proposed to ensure the occurrence of the sliding motion in a given finite time for four different chaotic systems. In theory, it is proved that the sliding mode technique can realize fast convergence for four different chaotic systems in the finite time. Some criteria and corollaries are derived for finite-time combination-combination synchronization of four different chaotic systems. Numerical simulation results are shown to verify the effectiveness and correctness of the combination-combination synchronization.     

Synchronization of N different coupled chaotic systems with ring and chain connections

Synchronization of N different coupled chaotic systems with ring and chain connections is investigated. The New system, the Chen system, the Lii system, the Lorenz system, and the Rossler system are used as examples in verifying effectiveness of the method. Based on the Lyapunov stability theory, the form of the controller is designed and the area of the coupling coefficients is determined. Simulations indicate that global synchronization of the N different chaotic systems can be realized by choosing appropriate coupling coefficients by using the controller.     

Shape synchronization of drive-response for a class of two-dimensional chaotic systems via continuous controllers

This paper investigates the drive-response synchronization in shape for a class of two-dimensional continuous systems of chaos. The shape of the chaotic attractor of the drive chaotic system is considered in this paper. Using the signed curvatures of plane curves to describe the shapes of trajectories for drive and response systems, the continuous controller for shape synchronization is synthesized based on the fundamental theorem on plane curves in classical differential geometry. The continuous controller synthesized can guarantee that the response system is synchronized with the drive chaotic system in shape. The shape synchronization is obtained in spite of different dimensions in drive and response systems. Finally, the Duffing oscillator is utilized as an illustrative example. Simulation results show that the method proposed in this paper is effective for the application of secure communication.     

A kind of binary scaling function projective lag synchronization of chaotic systems with stochastic perturbation

This paper investigates a kind of binary function projective lag synchronization of uncertain chaotic systems with stochastic perturbation. In comparison with those of the existing scaling function synchronization, we assume that the given scaling function can be the binary boundary function, even an n-ary boundary function. Based on the LaSalle-type invariance principle for stochastic differential equation, the adaptive control law is derived to make the state of two chaotic systems function projective lag synchronized. Some numerical is also given to show the effectiveness of the proposed method.