Controlled synchronization of discrete-time chaotic systems under communication constraints

This paper investigates the controlled synchronization problem for a class of nonlinear discrete-time chaotic systems subject to limited communication capacity. A general chaotic master system and its slave system with a controller are connected via a limited capacity channel. In this case, the effect of quantization errors is considered. A practical quantized scheme is proposed so that the synchronization error is input-to-state stable with respect to the transmission error. Meanwhile, the transmission error decays to zero exponentially. This implies that the synchronization error converges to zero under a limited communication channel. A?simulation example for the Fold chaotic system is presented to illustrate the effectiveness of the proposed method.     

Practical synchronization of second-order nonautonomous systems with parameter mismatch and its applications

This paper is concerned with robust practical synchronization for general second-order nonautonomous systems with parameter mismatch. Some simple yet general algebraic criteria are derived based on practical stability theory of nonautonomous dynamical systems. A?distinctive feature of this work is that the parameter mismatch cannot only be existed in system parameters, but also in external excitation ones. Furthermore, the obtained results are applied to a typical horizontal platform system and the representative forced Van der Pol oscillator. Subsequently, numerical simulations demonstrate the effectiveness of the criteria and the robustness of the control technique.     

Finite-time real combination synchronization of three complex-variable chaotic systems with unknown parameters via sliding mode control

The problem of real combination synchronization between three complex-variable chaotic systems with unknown parameters is investigated by nonsingular terminal sliding mode control in a finite time. Based on the adaptive laws and finite-time stability theory, a nonsingular terminal sliding mode control is designed to ensure the real combination synchronization of three complex-variable chaotic systems in a given finite time. It is theoretically gained that the introduced sliding mode technique has finite-time convergence and stability in both arriving and sliding mode phases. Numerical simulation results are given to show the effectiveness and reliability of the finite-time real combination synchronization.     

The theorem of the stability of linear nonautonomous systems under the frequently-acting perturbation

In the paper a theorem on the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of Malkin’s Theorem.     

Generalized reduced-order hybrid combination synchronization of three Josephson junctions via backstepping technique

In this paper, active backstepping design technique is applied to achieve reduced-order hybrid combination synchronization and reduced-order projective hybrid combination synchronization of three chaotic systems consisting of: (i) two third-order chaotic Josephson junctions as drives and one second-order chaotic Josephson junction as response system; (ii) one third-order chaotic Josephson junction as the drive and two second-order chaotic Josephson junctions as the slaves. Numerical simulations are performed to verify the feasibility and effectiveness of the analytical results. Reduced-order combination synchronization has more valuable practical applications to information processing in physical, biological, and social systems than the normal one master system and one slave system synchronization scheme.     

Control and synchronization of chaos systems using time-delay estimation and supervising switching control

In this paper, we present a new technique, developed using time-delay estimation (TDE) and supervising switching control (SSC), for the control and synchronization of chaos systems. The proposed technique consists of three units: a time-delay estimation unit that cancels system dynamics, a pole placement control unit that shapes error dynamics, and an SSC unit that is activated when the system dynamics are rapidly changing. We prove the stability of the closed-loop system using the Lyapunov analysis method. To verify the control and synchronization performance of the proposed technique (TDE-SSC), we compare it with TDC using numerical simulation. Our results indicate that the proposed scheme is an easily understood, numerically efficient, robust, and accurate solution for the control and synchronization of chaos systems.     

Modeling phase synchronization in systems of two and three coupled oscillators

We determine regions of synchronization of two and three globally coupled oscillators and describe the main mechanisms and bifurcations through which the synchronization of systems is lost.Published in Neliniini Kolyvannya, Vol. 7, No. 3, pp. 311–327, July–September, 2004.     

THE THEOREM OF THE STABILITY OF NONLINEAR NONAUTONOMOUS SYSTEMS UNDER THE FREQUENTLY-ACTING PERTURBATION—LIAPUNOV’S INDIRECT METHOD

In this paper the stability of nonlinear nonautonomous systems under the frequently-acting perturbation is studied. This study is a forward development of the study of the stability in the Liapunov sense; furthermore, it is of significance in practice since perturbations are often not single in the time domain. Malkin proved a general theorem about thesubject. To apply the theorem, however, the user has to construct a Liapunov function which satisfies specified conditions and it is difficult to find such a function for nonlinear nonautonomous systems. In the light of the principle of Liapunov’s indirect method, which is an effective method to decide the stability of nonlinear systems in the Liapunov sense, the authors have achieved several important conclusions expressed in the form of theorems to determine the stability of nonlinear nonautonomous systems under the frequently-acting perturbation.     

New globally asymptotical synchronization of chaotic systems under sampled-data controller

This paper investigates the robust synchronization problem of chaotic Lur’e systems with external disturbance using sampled-data \(H_{\infty }\) controller. The new method is based on a novel construction of piecewise differentiable Lyapunov–Krasovskii functional (LKF) in the framework of an input delay approach. Compared with existing works, the new LKF makes full use of the information on the nonlinear part of the system and introduces the novel terms, which guarantees the positive of the whole LKF. The output feedback \(H_{\infty }\) synchronization controller is presented to not only guarantee stable synchronization, but also reduce the effect of external disturbance to an \(H_{\infty }\) norm constraint. The proposed controller can be obtained by solving the linear matrix inequality problem. The effectiveness of the proposed method is demonstrated by the numerical simulations of Chua’s circuit.     

Combination synchronization of three different order nonlinear systems using active backstepping design

In this paper, two kinds of combination synchronization between two drive systems and one response system are investigated using active backstepping design. Firstly, increased-order combination synchronization between Lorenz system, Rössler system and hyperchaotic Lü system is considered. Secondly, reduced-order combination synchronization between hyperchaotic Lorenz system, hyperchaotic Chen system and Lü system is considered. According to Lyapunov stability theory and active backstepping design method, the corresponding controllers are both designed. Finally, several numerical examples are provided to illustrate the obtained results.     

Finite-time chaos control and synchronization of fractional-order nonautonomous chaotic (hyperchaotic) systems using fractional nonsingular terminal sliding mode technique

In this paper, a novel fractional-order terminal sliding mode control approach is introduced to control/synchronize chaos of fractional-order nonautonomous chaotic/hyperchaotic systems in a given finite time. The effects of model uncertainties and external disturbances are fully taken into account. First, a novel fractional nonsingular terminal sliding surface is proposed and its finite-time convergence to zero is analytically proved. Then an appropriate robust fractional sliding mode control law is proposed to ensure the occurrence of the sliding motion in a given finite time. The fractional version of the Lyapunov stability is used to prove the finite-time existence of the sliding motion. The proposed control scheme is applied to control/synchronize chaos of autonomous/nonautonomous fractional-order chaotic/hyperchaotic systems in the presence of both model uncertainties and external disturbances. Two illustrative examples are presented to show the efficiency and applicability of the proposed finite-time control strategy. It is worth to notice that the proposed fractional nonsingular terminal sliding mode control approach can be applied to control a broad range of nonlinear autonomous/nonautonomous fractional-order dynamical systems in finite time.     

Distributed adaptive control for multiple under-actuated Lagrangian systems under fixed or switching topology

Nonlinear Dynamics - This paper presents distributed adaptive tracking control algorithms under fixed or switching communication graph for a team of agents, each of which is a special class of...     

Reliability of a class of nonlinear systems under switching random excitations

Systems subjected to switching random excitations are practically significant because they include many safety-critical systems such as power plants and communication networks. In this paper, the reliability of multi-degree, nonlinear, non-integrable Hamiltonian systems subjected to switching random excitations is investigated. Such a system is formulated as a continuous–discrete hybrid based upon the Markov jump theory. Stochastic averaging is applied to suppress the rapidly varying parameters of the Markov jump process in order to generate a probability-weighted diffusion equation. The associated backward Kolmogorov equation is then set up, from which the approximate reliability function and probability density of first passage time are obtained. The utility and accuracy of this approximate procedure are demonstrated by two examples.     

The theorem of the stability of linear nonautonomous systems under the frequently-acting perturbation and its application in the stability analysis of robot

The necessary and sufficient condition of the stability of linear nonautonomous system under the frequently-acting perturbation has been given and proved on the basis of[1]and[2],and the theorem of the equivalence on the uniform and asymptotical stability in the sense of Liapunov and the stability under the frequently-acting perturbation of linear nonautonomous system has been given in this paper.Besides,the analysis of the dynamic stability of robot has been presented by applying the theorem in this paper,which is closer to reality.