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.     

Function projective synchronization of impulsive neural networks with mixed time-varying delays

In this paper, the exponential function projective synchronization of impulsive neural networks with mixed time-varying delays is investigated. Based on the contradiction method and analysis technique, some novel criteria are obtained to guarantee the function projective synchronization of considered networks via combining open-loop control and linear feedback control. As some special cases, several control strategies are given to ensure the realization of complete synchronization, anti-synchronization, and the stabilization of the addressed neural networks. Finally, two examples and their numerical simulations are given to show the effectiveness and feasibility of the proposed synchronization schemes.     

Exponential synchronization of chaotic neural networks: a matrix measure approach

Based on matrix measure and Halanay inequality, exponential synchronization of a class of chaotic neural networks with time-varying delays is investigated. Without constructing Lyapunov function, some simple but generic criteria for exponential synchronization of chaotic neural networks are derived. It is shown that the obtained results are easy to verify and simple to implement in practice. Two examples are given to illustrate the effectiveness of the presented synchronization scheme.     

Nonlinear synchronization on connected undirected networks

In this paper, we give sufficient conditions to have a complete synchronization of oscillators in connected undirected networks. The oscillators we are considering are not necessarily identical and the synchronization terms can be nonlinear. Many results in the literature deal with sufficient conditions insuring complete synchronization. This is a difficult problem since such conditions require to take into account the individual dynamics of the oscillators and also the graph topology. In this paper, we extend one of these results, the connection graph stability method, to nonlinear coupling functions and we also give an existence condition of trajectories of the oscillators. The sufficient conditions for synchronization presented in this paper are based on the study of a Lyapunov function and on the use of pseudometrics which enable us to link network dynamics and graph theory. These results are applied to a network of Chua’s oscillators and lead to an explicit condition insuring the complete synchronization of the oscillators.     

Almost sure cluster synchronization of Markovian switching complex networks with stochastic noise via decentralized adaptive pinning control

This paper investigates the issue of almost sure cluster synchronization in nonlinearly coupled complex networks with nonidentical nodes and time-varying delay. These networks are modulated by a continuous-time Markov chain and disturbed by a Brownian movement. The decentralized adaptive update law and pinning control protocol are employed in designing controllers for guaranteeing almost sure cluster synchronization. By constructing a novel stochastic Lyapunov–Krasovskii function and using the stochastic Lasalle-type invariance theorem, some sufficient conditions for almost sure cluster synchronization of the networks are derived. Finally, a numerical example is given to testify the effectiveness of the theoretical results.     

Observer-based impulsive chaotic synchronization of discrete-time switched systems

This paper investigates impulsive chaotic synchronization of discrete-time switched systems with state-dependent switching strategy. The parameter-dependent Lyapunov function (PDLF) technique is used to establish stability criteria for a class of switched systems consisting of both stable and unstable subsystems. With these criteria, sufficient conditions are given to achieve observer-based impulsive chaotic synchronization. Examples are presented to illustrate the criteria.     

Synchronization and chaos control by quorum sensing mechanism

Diverse rhythms are generated by thousands of oscillators that somehow manage to operate synchronously. By using mathematical and computational modeling, we consider the synchronization and chaos control among chaotic oscillators coupled indirectly but through a quorum sensing mechanism. Some sufficient criteria for synchronization under quorum sensing are given based on traditional Lyapunov function method. The Melnikov function method is used to theoretically explain how to suppress chaotic Lorenz systems to different types of periodic oscillators in quorum sensing mechanics. Numerical studies for classical Lorenz and Rössler systems illustrate the theoretical results.     

Fractional-order complex T system: bifurcations,chaos control,and synchronization

In this paper, the fractional-order complex T system is proposed. The dynamics of the system including symmetry, the stability of equilibrium points, bifurcations with variation of system parameters, and derivative orders are investigated. Period-doubling and tangent bifurcations with appropriate derivative orders and system parameters are observed. Besides, the control problem of the system is examined by using the feedback control technique. Furthermore, based on the stability theory of fractional-order systems, the scheme of function projective synchronization for the fractional-order complex T system is presented. The function projective synchronization for the system is realized by designing an appropriate synchronization controller. Numerical simulations are used to demonstrate the effectiveness and feasibility of the proposed scheme.     

A new Lyapunov approach for global synchronization of non-autonomous chaotic systems

This paper proposes a new approach for finding the Lyapunov function to study the sufficient global synchronization criterion of master-slave non-autonomous chaotic systems via linear state error feedback control. The approach is first demonstrated in a synchronization scheme for the second-order non-autonomous chaotic systems and then generalized to the schemes for the nth-order non-autonomous chaotic systems. Some algebraic synchronization criteria for the second-order chaotic systems are obtained. The sharpness of the new criteria is compared with that of the existing criteria of the same type by numerical examples.     

Parameter identification and synchronization of spatiotemporal chaos in uncertain complex network

Synchronization and parameter identification of a unidirectional star-network constructed by discrete spatiotemporal chaos systems with unknown parameters are studied. The synchronization principle of the network and design method of parameter recognition law are introduced. The function to be determined in the parameter recognition law and the range of adjusting parameter are obtained based on Lyapunov stability theory. Not only global synchronization of the network is realized, but also the unknown parameters in spatiotemporal chaos systems at the nodes of the network are identified. Discrete laser spatiotemporal chaos model is taken as each node of the network, and simulation results show the effectiveness of the synchronization principle and parameter recognition law.     

The complete synchronization condition in a network of piezoelectric micro-beams

This work deals with the dynamics of a network of piezoelectric micro-beams (a stack of disks). The complete synchronization condition for this class of chaotic nonlinear electromechanical system with nearest-neighbor diffusive coupling is studied. The nonlinearities within the devices studied here are in both the electrical and mechanical components. The investigation is made for the case of a large number of coupled discrete piezoelectric disks. The problem of chaos synchronization is described and converted into the analysis of the stability of the system via its differential equations. We show that the complete synchronization of N identical coupled nonlinear chaotic systems having shift invariant coupling schemes can be calculated from the synchronization of two of them. According to analytical, semi-analytical predictions and numerical calculations, the transition boundaries for chaos synchronization state in the coupled system are determined as a function of the increasing number of oscillators.

     

Modified projective and modified function projective synchronization of a class of real nonlinear systems and a class of complex nonlinear systems

The modified projective and modified function projective synchronization of a class of chaotic real nonlinear systems, or a class of chaotic complex nonlinear systems, have been widely reported in the previous studies, respectively. In the paper, the modified projective and modified function projective synchronization between a class of chaotic real nonlinear systems and a class of chaotic complex nonlinear systems are first investigated. Based on the Lyapunov stability theory, the drive real system and response complex system can be synchronized up to the desired scaling constants and functions, respectively. The corresponding numerical simulations are performed to verify and demonstrate the validity and feasibility of the presented idea.     

Projective-dual synchronization in delay dynamical systems with time-varying coupling delay

The existence of projective-dual-anticipating, projective-dual, and projective-dual-lag synchronization in a coupled time-delayed systems with modulated delay time is investigated via nonlinear observer design approach. Transition from projective-dual-anticipating to projective-dual synchronization and from projective-dual to projective-dual-lag synchronization as a function of variable coupling delay τ _p (t) is discussed. Using Krasovskii–Lyapunov stability theory, a general condition for projective-dual synchronization is derived. Numerical simulations on the chaotic Ikeda and Mackey–Glass systems are given to demonstrate the effectiveness of the theoretical results.     

Synchronization of PWL function-based 2D and 3D multi-scroll chaotic systems

We study the synchronization of a piecewise linear function-based chaotic system. That system generates multiple scrolls in multiple directions (two- and three-directions) on phase space. In this scenario, the design of a controller based on Generalized Hamiltonian forms is possible. As function of control signals, we propose a master?Cslave synchronization scheme using 2 ⁿ ?1 combinations to drive a nonlinear state observer. Associated with this, the piecewise linear functions of the slave are directly controlled by the state-variables of the master system. We computed the synchronization error for each combinations. Besides, the circuit synthesis based on operational amplifiers validates our synchronization scheme by means of SPICE simulations. We observed that the synchronization error at circuit level depends on the number of the control signals used. Our numerical and SPICE simulation results are in agreement showing the usefulness of the proposed approach.     

Study on spatiotemporal chaos synchronization among complex networks with diverse structures

The spatiotemporal chaos synchronization among complex networks with diverse structures is investigated. The spatiotemporal chaos systems are taken as the nodes of networks and constructed as some networks with diverse structures. The conditions of global synchronization among networks and the coupling function to be determined among diverse networks are discussed and confirmed based on stability theory. The Burgers equation with many practice physics processes, such as turbulent flow and heat-transfer, is adopted for example to imitate the experiment. It is found that the synchronization performance among all networks is very stable.     

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.     

Q–S synchronization between chaotic systems with double scaling functions

A double function Q–S synchronization (DFQSS) scheme of non-identical chaotic systems is proposed and analyzed with the assumption that all of the parameters are unknown. The sufficient conditions for achieving the double function Q–S synchronization with the desired scaling functions of two different chaotic systems (including the systems of non-identical dimension) are derived based on Lyapunov stability theory. By the adaptive control technique, the control laws and the corresponding parameter update laws are presented such that the DFQSS of non-identical chaotic systems is to be achieved. Numerical simulations and a brief discussion conclude the paper.     

Complete synchronization of chaotic complex nonlinear systems with uncertain parameters

Our main objective in this work is to investigate complete synchronization (CS) of n-dimensional chaotic complex systems with uncertain parameters. An adaptive control scheme is designed to study the synchronization of chaotic attractors of these systems. We applied this scheme, as an example, to study complete synchronization of chaotic attractors of two identical complex Lorenz systems. The adaptive control functions and the parameters estimation laws are calculated analytically based on the complex Lyapunov function. We show that the error dynamical systems are globally stable. Numerical simulations are computed to check the analytical expressions of adaptive controllers.     

Complex modified projective synchronization of two chaotic complex nonlinear systems

In this paper, we present a novel type of synchronization called complex modified projective synchronization (CMPS) and study it to a system of two chaotic complex nonlinear 3-dimensional flows, possessing chaotic attractors. Based on the Lyapunov function approach, a scheme is designed to achieve CMPS for such pairs of (either identical or different) complex systems. Analytical expressions for the complex control functions are derived using this scheme to achieve CMPS. This type of complex synchronization is considered as a generalization of several kinds of synchronization that have appeared in the recent literature. The master and slave chaotic complex systems achieved CMPS can be synchronized through the use of a complex scale matrix. The effectiveness of the obtained results is illustrated by a studying two examples of such coupled chaotic attractors in the complex domain. Numerical results are plotted to show the rapid convergence of modulus errors to zero, thus demonstrating that CMPS is efficiently achieved.     

A new chaotic attractor and its robust function projective synchronization

We introduce a simple chaotic system that contains one multiplier and one quadratic term. The system is similar to the generalized Lorenz system but is not topologically equivalent. The properties of the proposed chaotic system are examined by theoretical and numerical analysis. An analog chaotic circuit is implemented that realizes the chaotic system for the verification of its attractor. Furthermore, we propose a robust function projective synchronization using time delay estimation. A numerical simulation of synchronization between the proposed system and the Lorenz system demonstrates that the proposed approach provides fast and robust synchronization even in the presence of unknown parameter variations and disturbances.