In this paper, we give a framework for synchronization of dynamical systems which unifies many results in synchronization and control of dynamical systems, in particular chaotic systems. We define concepts such as asymptotical synchronization, partial synchronization and synchronization error bounds. We show how asymptotical synchronization is related to asymptotical stability. The main tool we use to prove asymptotical stability and synchronization is Lyapunov stability theory. We illustrate how many previous results on synchronization and control of chaotic systems can be derived from this framework. We will also give a characterization of robustness of synchronization and show that master-slave asymptotical synchronization in Chua’s oscillator is robust.
In this paper, we provide a scheme for synthesizing synchronized circuits and systems. Synchronization of the drive and response system is proved trivially without the need for computing numerically the conditional Lyapunov exponents. We give a definition of the driving and response system having the same functional form, which is more general than the concept of homogeneous driving by Pecora & Carroll [1991]. Finally, we show how synchronization coupled with chaos can be used to implement secure communication systems. This is illustrated with examples of secure communication systems which are inherently error-free in contrast to the signal-masking schemes proposed in Cuomo & Oppenheim [1993a,b] and Kocarev et al. [1992].
We study synchronization in an array of coupled identical nonlinear dynamical systems where the coupling topology is expressed as a directed graph and give synchronization criteria related to properties of a generalized Laplacian matrix of the directed graph. In particular, we extend recent results by showing that the array synchronizes for sufficiently large cooperative coupling if the underlying graph contains a spanning directed tree. This is an intuitive yet nontrivial result that can be paraphrased as follows: if there exists a dynamical system which influences directly or indirectly all other systems, then synchronization is possible for strong enough coupling. The converse is also true in general.
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