Synchronization and anti-synchronization of new uncertain fractional-order modified unified chaotic systems via novel active pinning control

This paper discusses the synchronization and anti-synchronization of new uncertain fractional-order unified chaotic systems (UFOUCS). Based on the idea of active control, a novel active pinning control strategy is presented, which only needs a state of new UFOUCS. The proposed controller can achieve synchronization between a response system and a drive system, and ensure the synchronized robust stability of new UFOUCS. Numerical simulations of new UFOUCS show that the controller can make fractional-order unified chaotic systems (FOUCS) achieve synchronization or anti-synchronization in a quite short period and both are of good robust stability.     

Synchronization and anti-synchronization of Lu and Bhalekar–Gejji chaotic systems using nonlinear active control

This paper aims at synchronization and anti-synchronization between Lu chaotic system, a member of unified chaotic system, and recently developed Bhalekar–Gejji chaotic system, a system which cannot be derived from the member of unified chaotic system. These synchronization and anti-synchronization have been achieved by using nonlinear active control since the parameters of both the systems are known. Lyapunov stability theory is used and required condition is derived to ensure the stability of error dynamics. Controller is designed by using the sum of relevant variables in chaotic systems. Simulation results suggest that proposed scheme is working satisfactorily.     

Synchronization and anti-synchronization of chaotic oscillators under input saturation

This paper addresses the design of simple state feedback controllers for synchronization and anti-synchronization of chaotic oscillators under input saturation and disturbance. By employing sector condition, linear matrix inequality (LMI)-based sufficient conditions are derived to design (global or local) controllers for chaos synchronization. The proposed local synchronization strategy guarantees a region of stability in terms of difference between states of the master–slave systems. This region of stability can be enlarged by means of an LMI-based optimization algorithm, through which asymptotic synchronization of chaotic oscillators can be ensured for a large difference in their initial conditions. Further, a novel LMI-based robust control strategy is developed, for local synchronization of input-constrained chaotic oscillators, by providing an upper bound on synchronization error in terms of disturbance and initial conditions of chaotic systems. Moreover, the proposed robust state feedback control methodology is modified to provide an inaugural treatment for robust anti-synchronization of chaotic systems under input saturation and disturbance. The results of the proposed methodologies are verified through numerical simulations for synchronization and anti-synchronization of the master–slave chaotic Chua’s circuits under input saturation.     

A new hyper-chaotic system and its synchronization

This paper presents a new hyper-chaotic system obtained by adding a nonlinear controller to the third equation of the three-dimensional autonomous Chen–Lee chaotic system. Computer simulations demonstrated the hyper-chaotic dynamic behaviors of the system. Numerical results revealed that the new hyper-chaotic system possesses two positive exponents. It was also found that the structure of the hyper-chaotic attractors is more complex than those of the Chen–Lee chaotic system. Furthermore, the hybrid projective synchronization (HPS) of the new hyper-chaotic systems was studied using a nonlinear feedback control. The nonlinear controller was designed according to Lyapunov’s direct method to guarantee HPS, which includes synchronization, anti-synchronization, and projective synchronization. Numerical examples are presented in order to illustrate HPS.     

Full state hybrid projective synchronization in continuous-time chaotic (hyperchaotic) systems

Chaos synchronization, as an important topic, has become an active research subject in nonlinear science. Over the past two decades, chaos synchronization between nonlinear systems has been extensively studied, and many types of synchronization have been announced. This paper introduces another novel type of chaos synchronization – full state hybrid projective synchronization (FSHPS), which includes complete synchronization, anti-synchronization and projective synchronization as its special item. Based on the Lyapunov’s direct method, the general FSHPS scheme is given and illustrated with Lorenz chaotic system and hyperchaotic Chen system as examples. Numerical simulations are used to verify the effectiveness of the proposed scheme.     

Synchronization and anti-synchronization coexist in Chen–Lee chaotic systems

This study demonstrates that synchronization and anti-synchronization can coexist in Chen–Lee chaotic systems by direct linear coupling. Based on Lyapunov’s direct method, a linear controller was designed to assure that two different types of synchronization can simultaneously be achieved. Further, the hybrid projective synchronization of Chen–Lee chaotic systems was studied using a nonlinear control scheme. The nonlinear controller was designed according to the Lyapunov stability theory to guarantee the hybrid projective synchronization, including synchronization, anti-synchronization, and projective synchronization. Finally, numerical examples are presented in order to illustrate the proposed synchronization approach.     

Synchronization of chaotic systems with parameter driven by a chaotic signal

Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.     

Synchronization of chaotic systems with parameter driven by a chaotic signal

Chaos control with driving parameter scheme in uncoupled identical chaotic oscillators is presented. By driving the parameter of chaotic systems using external chaotic signal, synchronization and anti-synchronization can be implemented. Numerical simulations show that either synchronization or anti-synchronization can appear depending significantly on initial condition and on driving strength. The proposed method is particularly suited for a variety of chaotic systems, which cannot couple with each other in engineering.     

Synchronization and anti-synchronization coexist in two-degree-of-freedom dissipative gyroscope with nonlinear inputs

This study demonstrates that synchronization and anti-synchronization can coexist in two-degree-of-freedom dissipative gyroscope system with input nonlinearity. Because of the nonlinear terms of the gyroscope system, the system exhibits complex motions containing regular and chaotic motions. Using the variable structure control technique, a novel control law is established which guarantees the hybrid projective synchronization including synchronization, anti-synchronization and projective synchronization even when the control input nonlinearity is present. By Lyapunov stability theory with control terms, two suitable sliding surfaces are proposed to ensure the stability of the controlled closed-loop system in sliding mode, and two variable structure controllers (VSC) are designed to guarantee the hitting of the sliding surfaces. Numerical simulations are presented to verify the proposed synchronization approach.     

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.     

Synchronization in pair-coupled maps with invariant measure

Chaos synchronization, as an important topic, has become an active research subject in non-linear science. By considering a symmetric two-dimensional map that possesses invariant measure in its diagonal and anti-diagonal invariant sub-manifolds, we have been able to introduce the most general pair-coupled map possessing invariant measure at synchronized or anti-synchronized states. Then chaotic synchronization and anti-synchronization are investigated in introduced model. We have calculated Kolmogrov–Sinai entropy and Lyapunov exponent as another tool to study the stability of pair-coupled map at synchronized and anti-synchronization states.     

Synchronization and anti-synchronization of a hyperchaotic Chen system

Based on the active control theory, synchronization and anti-synchronization between two identical chaotic systems is investigated. Anti-synchronization can be characterized by the vanishing of the sum of relevant variables. Through rigorous mathematical theory, the sufficient condition is drawn for the stability of the error dynamics, where the controllers are designed by using the sum of the relevant variables in chaotic systems. Numerical simulations are performed for Chen hyperchaotic dynamical system to demonstrate the effectiveness of the proposed control strategy.     

A new scheme of general hybrid projective complete dislocated synchronization

Based on the Lyapunov stability theorem, a new type of chaos synchronization, general hybrid projective complete dislocated synchronization (GHPCDS), is proposed under the framework of drive-response systems. The difference between the GHPCDS and complete synchronization is that every state variable of drive system does not equal the corresponding state variable, but equal other ones of response system while evolving in time. The GHPCDS includes complete dislocated synchronization, dislocated anti-synchronization and projective dislocated synchronization as its special item. As examples, the Lorenz chaotic system, Rössler chaotic system, hyperchaotic Chen system and hyperchaotic Lü system are discussed. Numerical simulations are given to show the effectiveness of these methods.     

Generalized synchronization of strictly different systems: Partial-state synchrony

Generalized synchronization (GS) occurs when the states of one system, through a functional mapping are equal to states of another. Since for many physical systems only some state variables are observable, it seems convenient to extend the theoretical framework of synchronization to consider such situations. In this contribution, we investigate two variants of GS which appear between strictly different chaotic systems. We consider that for both the drive and response systems only one observable is available. For the case when both systems can be taken to a complete triangular form, a GS can be achieved where the functional mapping between drive and response is found directly from their Lie-algebra based transformations. Then, for systems that have dynamics associated to uncontrolled and unobservable states, called internal dynamics, where only a partial triangular form is possible via coordinate transformations, for this situation, a GS is achieved for which the coordinate transformations describe the functional mapping of only a few state variables. As such, we propose definitions for complete and partial-state GS. These particular forms of GS are illustrated with numerical simulations of well-known chaotic benchmark systems.     

Complete synchronization,phase synchronization and parameters estimation in a realistic chaotic system

The two-parameter phase space in certain nonlinear system is investigated and the chaotic region of parameters are measured to show its chaotic properties. Within the chaotic parameter region, the complete synchronization, phase synchronization and parameters estimation are discussed in detail by using adaptive synchronization scheme and Lyapunov stability theory. Two changeable gain coefficients are introduced into the controllable positive Lyapunov function and thus the parameter observers. It is found that complete synchronization or phase synchronization occurs with different controllers being used though the parameter observers are the same. Phase synchronization is observed when zero eigenvalue of Jacobi matrix, which is composed of the errors of corresponding variables in the drive and driven chaotic systems. The optimized selection of controllers can induce transition of phase synchronization and complete synchronization.     

Switching among three different kinds of synchronization for delay chaotic systems

We found that the complete synchronization, anticipating synchronization and lag synchronization can be reached by the same kind of one way coupling for a large class of chaotic delay system. By changing the transformation time of the coupling signal we can switch from anticipating synchronization to complete synchronization, and then to lag synchronization. Numerical simulation for three chaotic delay systems were presented, one of them was novel which had two degree of freedoms, and the other two were the well known Ikeda and Mackey–Glass system which are one degree of freedom chaotic delay system. The theoretical analysis and the numerical simulation agreed perfect good.     

Anti-synchronization of uncertain unified chaotic systems with dead-zone nonlinearity

This paper addresses chaos anti-synchronization of uncertain unified chaotic systems with dead-zone input nonlinearity. Using the sliding mode control technique and Lyapunov stability theory, a proportional–integral (PI) switching surface is proposed to ensure the stability of the closed-loop error system in sliding mode. Then a sliding mode controller (SMC) is proposed to guarantee the hitting of the switching surface even with uncertainties and the control input containing dead-zone nonlinearity. Some simulation results are included to demonstrate the effectiveness and feasibility of the proposed synchronization scheme.     

The function cascade synchronization method and applications

In this paper, a new function cascade synchronization method of chaos system is proposed to achieve generalized projective synchronization for chaotic systems. Based on Laypunov stability, the proposed synchronization technique is applied to three famous chaotic systems: the unified chaotic system, Liu system and Rössler system, which can make the states of two identical chaotic systems asymptotically synchronized by choosing different special suitable error functions. Numerical simulations are presented to show the effectiveness.     

Adaptive projective synchronization for a class of switched chaotic systems

This paper addresses the problem of projective synchronization of chaotic systems and switched chaotic systems by adaptive control methods. First, a necessary and sufficient condition is proposed to show how many state variables can realize projective synchronization under a linear feedback controller for the chaotic systems. Then, accordingly, a new algorithm is given to select all state variables that can realize projective synchronization. Furthermore, according to the results of the projective synchronization of chaotic systems, the problem of projective synchronization of the switched chaotic systems comprised by the unified chaotic systems is investigated, and an adaptive global linear feedback controller with only one input channel is designed, which can realize the projective synchronization under the arbitrary switching law. It is worth mentioning that the proposed method can also realize complete synchronization of the switched chaotic systems. Finally, the numerical simulation results verify the correctness and effectiveness of the proposed method.     

On the synchronization of different chaotic oscillators

The synchronization of two different chaotic oscillators is studied, based on an open-loop control – the entrainment control. We consider two types of synchronization: complete synchronization and effectively complete synchronization. The sufficient conditions that two different systems can be synchronized by this method is discussed. Furthermore, a hierarchical idea to synchronize multiple chaotic subsystems is proposed.