Chaotic synchronization based on stability criterion of linear systems

This work is concerned with anti-synchronization of Liu system and Lorenz system. Based on Lyapunov stability theory, different controllers are designed to anti-synchronize the two non-identical chaotic systems, active control is used when parameters are known, while the adaptive control law and the parameter update rule are derived via adaptive control when parameters are uncertain. Moreover, the convergence speeds of the scheme can be adjusted by changing the control coefficients. Finally, numerical simulations are also shown to verify the results.

Section: Introductionmentioning

confidence: 99%

Anti-synchronization of Liu system and Lorenz system with known or unknown parameters

Wang

Shi

2008

“…It has been observed that different synchronisation levels may be achieved between interacting dynamical systems [3]. In complete (or identical) synchronisation, trajectories of equivalent state variables are coincident [17,36]. This kind of synchronisation was firstly shown for two identical chaotic systems unidirectionally coupled [26].…”

Section: Introductionmentioning

confidence: 95%

Dynamical behaviour of multi-particle large-scale systems

Lopes

Machado

2012

Collective behaviours can be observed in both natural and man-made systems composed of a large number of elemental subsystems. Typically, each elemental subsystem has its own dynamics but, whenever interaction between individuals occurs, the individual behaviours tend to be relaxed, and collective behaviours emerge. In this paper, the collective behaviour of a large-scale system composed of several coupled elemental particles is analysed. The dynamics of the particles are governed by the same type of equations but having different parameter values and initial conditions. Coupling between particles is based on statistical feedback, which means that each particle is affected by the average behaviour of its neighbours. It is shown that the global system may unveil several types of collective behaviours, corresponding to partial synchronisation, characterised by the existence of several clusters of synchronised subsystems, and global synchronisation between particles, where all the elemental particles synchronise completely.

“…It is precisely because so many uncertainties and intrinsic nonlinearities, eliminating or weakening the nonlinear terms in error dynamical systems is essential. In some more extreme cases, another control scheme has to be involved to eliminate all of the nonlinear terms and readjust the structure of constant matrix [21]. Due to the simple configuration and easy implementation, the unidirectional and bidirectional linear error feedback coupling schemes were adopted to control and synchronize chaotic and hyper-chaotic systems [22][23][24][25][26][27].…”

Section: Introductionmentioning

confidence: 99%

“…From rigorously mathematical theory, some sufficient conditions of global synchronization for linearly coupled chaotic systems are presented in Lü's paper [26]. However, most of their works have a common problem, that is, the chaotic systems they considered must be able to be decomposed into their linear and nonlinear parts independently [21][22][23][24][25]27,32]. Specifically, the n-dimensional chaotic systems must be able to be rewritten in the form of…”

Section: Introductionmentioning

confidence: 99%

Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling

Leung

Chu

et al. 2015

Based on the stability theory of fractionalorder system, a novel unidirectional adaptive full-state linear error feedback coupling scheme is extended to control and synchronize all of fractional-order differential (FOD) chaotic systems with in-commensurate (and commensurate) orders. The feedback strength is adaptive to an updated law rather than prescribed as a constant. The convergence speed of feedback strength is regulated by a constant. With rigorous linear algebraic theorems and precisely numerical matrix computations, a reasonable interval in which the ultimate final control strength dwells is suggested, and the reliability of synchronization state is guaranteed. It demonstrates that the unidirectional full-state linear feedback coupling scheme can be adopted to control and synchronize FOD chaotic systems directly. Numerical simulations of three representative FOD chaotic systems illustrate the effectiveness of the proposed scheme.