Adaptive generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimensions

Generalized function matrix projective lag synchronization of uncertain complex dynamical networks with different dimension of nodes via adaptive control method is investigated in this paper. Based on Lyapunov stability theory, adaptive controller is obtained and unknown parameters of both the drive network and the response network are estimated by adaptive laws. In addition, the three-dimension chaotic system and the four-dimension hyperchaotic system, respectively, as the nodes of the drive and response network are analyzed in detail, and numerical simulation results are presented to illustrate the effectiveness of the theoretical results.     

Adaptive impulsive synchronization of uncertain drive-response complex-variable chaotic systems

In this paper, impulsive synchronization of drive-response complex-variable chaotic systems is investigated. The drive-response systems with known parameters is considered via impulsive control and adaptive scheme as well as systems with unknown parameters. Noticeably, adaptive strategy is adopted to relax the restriction on the impulsive interval, and the system parameters need not to be known beforehand. According to the Lyapunov stability theory, some synchronization criteria are derived and verified by several numerical simulations.     

Complex projective synchronization in coupled chaotic complex dynamical systems

In previous papers, the projective factors are always chosen as real numbers, real matrices, or even real-valued functions, which means the coupled systems evolve in the same or inverse direction simultaneously. However, in many practical situations, the drive-response systems may evolve in different directions with a constant intersection angle. Therefore, the projective synchronization with respect to a complex factor, called complex projective synchronization (CPS), should be taken into consideration. In this paper, based on Lyapunov stability theory, three typical chaotic complex dynamical systems are considered and the corresponding controllers are designed to achieve the complex projective synchronization. Further, an adaptive control method is adopted to design a universal controller for partially linear systems. Numerical examples are provided to show the effectiveness of the proposed method.     

Adaptive modified function projective synchronization and parameter identification of uncertain hyperchaotic (chaotic) systems with identical or non-identical structures

Modified function projective synchronization (MFPS), which generalizes many kinds of synchronization form, has received great attention recently. Based on the active control method and adaptive control technique, a general formula for designing the controllers is proposed to achieve adaptive MFPS, which corrects several incomplete results that have been reported recently. In addition, this paper derives the sufficient condition for parameter identification, which was not mentioned in much of the relevant literature concerning MFPS. Furthermore, we extend the MFPS scheme to the cases that the drive and response systems come with non-identical structures. The proposed method is both theoretically rigorous and practically feasible, which has the merits that it can not only achieve the full-state MFPS but also identify the fully unknown parameters in the synchronization process. The theoretical results are successfully applied to three typical illustrative cases: the adaptive MFPS of two identical 4-D hyperchaotic systems with unknown parameters in the response system, the adaptive MFPS between a 5-D hyperchaotic system and a 4-D hyperchaotic system with unknown parameters in the drive system and the adaptive MFPS between a 3-D chaotic system and a 4-D hyperchaotic system when the parameters in the drive system and response system are all unknown. For each case the controller functions and parameter update laws are well designed in detail. Moreover, the corresponding numerical simulations are presented, which agree well with the theoretical analysis.     

Modified projective synchronization of stochastic fractional order chaotic systems with uncertain parameters

In this paper, the synchronization of fractional order chaotic systems with random and uncertain parameters is analyzed. Firstly, based on the orthogonal polynomial expansion, the fractional order Lü and Lorenz systems with random and uncertain parameters are reduced into the equivalent deterministic systems. Secondly, modified projective synchronization of equivalent deterministic Lü and Lorenz systems is explored. Lastly, the theoretical results are verified by the numerical simulations.     

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.     

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.     

Disturbance-observer-based robust synchronization control of uncertain chaotic systems

In this paper, a robust synchronization control scheme is proposed for chaotic systems in the presence of system uncertainties and unknown external disturbances. For the synchronization error system, the compound disturbance which is estimated using the disturbance observer cannot be directly measured. If the gain matrix is properly chosen, the disturbance observer can approximate the unknown compound disturbance well. And then, the constrained robust synchronization control scheme is presented for uncertain chaotic systems based on the output of disturbance observer. In the design of a robust synchronization control scheme, the effect of unknown control input constraint has been explicitly considered to guarantee the synchronization performance. Numerical simulation results are presented to illustrate the effectiveness of the proposed constrained synchronization control scheme for uncertain chaotic systems.     

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.     

Complex function projective synchronization of complex chaotic system and its applications in secure communication

To develop secure communication, the paper presents complex function projective synchronization (CFPS) of complex chaotic systems. Aimed to coupled complex chaotic system, the control law is derived to make the complex state vectors asymptotically synchronize up to a desired complex function matrix. Based on CFPS, a novel communication scheme is further designed in theory. Its main idea is chaotic masking in essence, but the transmitted signal is the derivative of the product of the information signal and chaotic signal. As the complex scaling functions are arbitrary and more unpredictable than real scaling functions, and the product and derivative operations of complex numbers are complicated, the possibility that an interceptor extracts the information from the transmitted signal is greatly reduced. The communication system can transmit analog signal and digital symbols with fast transmission and high security, especially low bit-error rate and the strong robustness to noise for digital symbols. The corresponding numerical simulations are performed to verify and illustrate the analytical results.     

Adaptive impulsive synchronization of nonlinear chaotic systems

In this paper, an adaptive impulse control with only one restriction criterion is derived to achieve synchronization of nonlinear chaotic systems in the exponential rate of convergence. It is assumed that the system satisfies the local Lipschitz condition and the Lipschitz constant is estimated by an augmented adaptation equation. The significance of the related control parameters in the criterion is also discussed in detail. The Duffing and the Lorenz systems have been simulated to illustrate the theoretical analysis.     

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.