Parameter identification and adaptive impulsive synchronization of uncertain complex-variable chaotic systems

Synchronization of nonlinear dynamical systems with complex variables has attracted much more attention in various fields of science and engineering. In this paper, the problem of parameter identification and adaptive impulsive synchronization for a class of chaotic (hyperchaotic) complex nonlinear systems with uncertain parameters is investigated. Based on the theories of adaptive control and impulsive control, a synchronization scheme is designed to make a class of chaotic and hyperchaotic complex systems asymptotically synchronized, and uncertain parameters are identified simultaneously in the process of synchronization. Particularly, the proposed adaptive–impulsive control laws for synchronization are simple and can be readily applied in practical applications. The synchronization of two identical chaotic complex Chen systems and two identical hyperchaotic complex Lü systems are taken as two examples to verify the feasibility and effectiveness of the proposed controllers and identifiers.     

Synchronization of nonlinear chaotic electromechanical gyrostat systems with uncertainties

This paper solves the problem of robust synchronization of nonlinear chaotic gyrostat systems in a given finite time. The parameters of both master and slave chaotic gyrostat systems are assumed to be unknown in advance. In addition, the gyrostat systems are disturbed by unknown model uncertainties and external disturbances. Suitable update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in finite time. The precise value of the convergence time is given. A numerical simulation demonstrates the applicability and efficiency of the proposed finite-time synchronization strategy.     

Adaptive control for electromechanical systems considering dead-zone phenomenon

The electromechanical gyrostat is a fourth-order nonautonomous system that exhibits very rich behavior such as chaos. In recent years, synchronization of nonautonomous chaotic systems has found many useful applications in nonlinear science and engineering fields. On the other hand, it is well known that the finite-time control techniques demonstrate good robustness and disturbance rejection properties. This paper studies the potential application of the finite-time control techniques for synchronization of nonautonomous chaotic electromechanical gyrostat systems in finite time. It is assumed that all the parameters of both drive and response systems are unknown parameters in advance. Moreover, the effects of dead-zone nonlinearities in the control inputs are also taken into account. Some adaptive controllers are introduced to synchronize two gyrostat systems in different scenarios within a given finite-time. Two illustrative examples are presented to demonstrate the efficiency and robustness of the proposed finite-time synchronization strategy.     

Analyzing projective synchronization on different scaling factors in a drive-response coupling dynamical network with time-varying delays

In this paper, projective synchronization of drive-response coupled dynamical network with delayed system nodes and coupling time-varying delay is investigated via impulsive control, where the scaling factors are different from each other. Different controllers are designed to achieve the projective synchronization: only impulsive control is used when the scaling factors need extra limitation, while an extra controller, that is, a simple linear feedback controller, is added when the scaling factors don??t need extra limitation. Based on the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization of such coupled network are established, and an estimate of the upper bound of impulsive intervals ensuring global exponential synchronization of drive-response coupled dynamical network is also given. Numerical examples on the time-delay Lorenz chaotic systems are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Nonlinear observer-based impulsive synchronization in chaotic systems with multiple attractors

The issue of impulsive synchronization of the coupled Newton–Leipnik system is investigated. Based on the impulsive stability theory, nonlinear observer-based impulsive synchronization scheme is derived. A new and less conservative criteria for impulsive synchronization via nonlinear observer is proposed. The boundary of the stable regions is also estimated. One important advantage of the proposed method is that it is also applicable for the systems with more than one attractor. Numerical simulations on Newton–Leipnik system are illustrated to verify the theoretical results.     

Synchronization analysis of directed complex networks with time-delayed dynamical nodes and impulsive effects

In this paper, the effect of impulses on the synchronization of a class of general delayed dynamical networks is analyzed. The network topology is assumed to be directed and weakly connected with a spanning tree. Two types of impulses occurred in the states of nodes are considered: (i) synchronizing impulses meaning that they can enhance the synchronization of dynamical networks; and (ii) desynchronizing impulses defined as the impulsive effects can suppress the synchronization of dynamical networks. For each type of impulses, some novel and less conservative globally exponential synchronization criteria are derived by using the concept of average impulsive interval and the comparison principle. It is shown that the derived criteria are closely related with impulse strengths, average impulsive interval, and topology structure of the networks. The obtained results not only can provide an effective impulsive control strategy to synchronize an arbitrary given delayed dynamical network even if the original network may be asynchronous itself but also indicate that under which impulsive perturbations globally exponential synchronization of the underlying delayed dynamical networks can be preserved. Numerical simulations are finally given to demonstrate the effectiveness of the theoretical results.     

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.     

Synchronization between fractional-order Ravinovich–Fabrikant and Lotka–Volterra systems

This article examines the synchronization performance between two fractional-order systems, viz., the Ravinovich?CFabrikant chaotic system as drive system and the Lotka?CVolterra system as response system. The chaotic attractors of the systems are found for fractional-order time derivatives described in Caputo sense. Numerical simulation results which are carried out using Adams?CBoshforth?CMoulton method show that the method is reliable and effective for synchronization of nonlinear dynamical evolutionary systems. Effects on synchronization time due to the presence of fractional-order derivative are the key features of the present article.     

Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization

The primary objective of this paper is to propose a new approach for analyzing pinning stability in a complex dynamical network via impulsive control. A?simple yet generic criterion of impulsive pinning synchronization for such coupled oscillator network is derived analytically. It is shown that a single impulsive controller can always pin a given complex dynamical network to a homogeneous solution. Subsequently, the theoretic result is applied to a small-world (SW) neuronal network comprised of the Hindmarsh?CRose oscillators. It turns out that the firing activities of a single neuron can induce synchronization of the underlying neuronal networks. This conclusion is obviously in consistence with empirical evidence from the biological experiments, which plays a significant role in neural signal encoding and transduction of information processing for neuronal activity. Finally, simulations are provided to demonstrate the practical nature of the theoretical results.     

Lag synchronization of a class of chaotic systems with unknown parameters

Research on chaos synchronization of dynamical systems has been largely reported in literature. However, synchronization of different structure—uncertain dynamical systems—has received less attention. This paper addresses synchronization of a class of time-delay chaotic systems containing uncertain parameters. A unified scheme is established for synchronization between two strictly different time-delay uncertain chaotic systems. The synchronization is successfully achieved by designing an adaptive controller with the estimates of the unknown parameters and the nonlinear feedback gain. The result is rigorously proved by the Lyapunov stability theorem. Moreover, we illustrate the application of the proposed scheme by numerical simulation, which demonstrates the effectiveness and feasibility of the proposed synchronization method.     

Mixed outer synchronization of dynamical networks with nonidentical nodes and output coupling

Outer synchronization between the drive network and the response network has attracted much more attention in various fields of science and engineering. In this paper, mixed outer synchronization between two complex dynamical networks with nonidentical nodes and output coupling is investigated via impulsive hybrid control, that is, an adaptive feedback controller with impulsive control effects. Moreover, both the cases of complex networks without and with coupling delay are considered. According to the stability analysis of the impulsive functional differential equation, several sufficient conditions for the networks to achieve mixed outer synchronization are derived. Numerical examples are presented finally to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Full- and reduced-order synchronization of a class of time-varying systems containing uncertainties

Investigation on chaos synchronization of autonomous dynamical systems has been largely reported in the literature. However, synchronization of time-varying, or nonautonomous, uncertain dynamical systems has received less attention. The present contribution addresses full- and reduced-order synchronization of a class of nonlinear time-varying chaotic systems containing uncertain parameters. A unified framework is established for both the full-order synchronization between two completely identical time-varying uncertain systems and the reduced-order synchronization between two strictly different time-varying uncertain systems. The synchronization is successfully achieved by adjusting the determined algorithms for the estimates of unknown parameters and the linear feedback gain, which is rigorously proved by means of the Lyapunov stability theorem for nonautonomous differential equations together with Barbalat’s lemma. Moreover, the synchronization result is robust against the disturbance of noise. We illustrate the applicability for full-order synchronization using two identical parametrically driven pendulum oscillators and for reduced-order synchronization using the parametrically driven second-order pendulum oscillator and an additionally driven third-order Rossler oscillator.     

Projective synchronization in a driven-response dynamical network with coupling time-varying delay

This paper studies the projective synchronization in a driven-response dynamical network with coupling time-varying delay model via impulsive control, in which the weights of links are time varying. Based on the stability analysis of the impulsive functional differential equations, some sufficient conditions for the projective synchronization are derived. Numerical simulations are provided to verify the correctness and effectiveness of the proposed method and results.     

Synchronization criteria for impulsive complex dynamical networks with time-varying delay

This paper investigates the output synchronization of a class of impulsive complex dynamical networks with time-varying delay. By constructing suitable Lyapunov functionals, some new and useful conditions are obtained to guarantee the local and global exponential output synchronization of the impulsive complex networks. Finally, numerical examples are given to demonstrate the effectiveness of the theoretical results.     

Adaptive finite-time stabilization of uncertain non-autonomous chaotic electromechanical gyrostat systems with unknown parameters

This paper deals with the problem of robust finite-time stabilization of non-autonomous chaotic gyrostat systems. It is assumed that the parameters of the gyrostat system are completely unknown in advance and the system is perturbed by unknown uncertainties and disturbances. Some update laws are proposed to estimate the unknown parameters. Based on the finite-time control idea and the update laws, appropriate control laws are designed to ensure the stabilization of the closed-loop system in a finite time. The finite-time stability and convergence of the closed-loop system are analytically proved. A numerical simulation is given to demonstrate the applicability and robustness of the proposed finite-time controller and to verify the theoretical results.     

Nonlinear state-observer control for projective synchronization of a fractional-order hyperchaotic system

In this paper, we consider an observer-based control approach for manipulating projective synchronization of nonlinear systems in high dimensional. Based on the stability theory of the fractional-order dynamical system, a nonlinear state observer is designed which can achieve projective synchronization in a class of high dimensional fractional-order hyperchaotic systems without restriction of partial-linearity and calculating the Lyapunov index of system. Simulation studies are included to demonstrate the effectiveness and feasibility of the proposed approach and synthesis procedures.     

Impulsive control induced effects on dynamics of single and coupled ODE systems

The dynamics of differential system can be changed very obviously after inputting impulse signals. Previous studies show that the single chaotic system can be controlled to periodic motions using impulsive control method. It was well known that the dynamics of hyper-chaotic and coupled systems are very important and more complex than those of a single system. In this paper, particular impulsive control of the hyper-chaotic Lü system was proposed, which is with outer impulsive signals. It can be seen that such impulsive strategy can generate chaos from periodic orbit or control chaos to periodic orbit etc. For the first time, impulsive control induced effects on dynamics of coupled systems are considered in this paper, where the impulse effect has outer input signals. Many interesting and useful results are obtained. The coupled system can realize synchronization and its synchronization manifold can be changed with such impulsive control signals. Strict theories are given, and numerical simulations confirm the correctness of theoretical results.     

Adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling

This paper investigates the adaptive-impulsive projective synchronization of drive-response delayed complex dynamical networks with time-varying coupling, in which the weights of links between two connected nodes are time varying. By the stability analysis of the impulsive functional differential equation, the sufficient conditions for achieving projective synchronization are obtained, and a hybrid controller, that is, an adaptive feedback controller with impulsive control effects is designed. The numerical examples are presented to illustrate the effectiveness and advantage of the proposed synchronization criteria.     

Outer synchronization of complex networks with delay via impulse

Synchronization between the driving network and the responding network (outer synchronization) has attracted increasing attention from various fields of science and engineering. In this paper, we address outer synchronization of complex networks with delays. Both the cases of coupling delay and node delay are considered. Employing the impulsive control method which is simple, efficient, low cost, and easy to implement in practical applications, we obtain some sufficient conditions of outer synchronization. It indicates that outer synchronization can be achieved if the maximal impulsive intervals are less than a critical value. Numerical simulations are also given to demonstrate the effectiveness of the proposed impulsive control scheme.     

Synchronization of chaotic systems using fuzzy impulsive control

This paper investigates the problem of fuzzy impulsive control to synchronize two chaotic systems using a novel time-dependent Lyapunov function approach. Compared with the existing time-independent Lyapunov methods, the proposed method enables us to exploit more information on the impulsive intervals. Initially, using the Lyapunov technique and two parameterized linear matrix inequality (LMI) techniques, some less conservative synchronization criteria via a fuzzy impulsive controller using the states of both drive and response chaotic systems are derived. Subsequently, an LMI approach to designing such a fuzzy impulsive controller is developed to realize the synchronization. Finally, the proposed method is applied to the chaotic Lorenz system and Rösler system to illustrate its effectiveness.