Generalized synchronization of chaos in directionally coupled chaotic systems

Abstract

Synchronization of chaotic systems is frequently taken to mean actual equality of the variables of the coupled systems as they evolve in time. We explore a generalization of this condition, which equates dynamical variables from one subsystem with a function of the variables of another subsystem. This means that synchronization implies a collapse of the overall evolution onto a subspace of the system attractor in full space. We explore this idea in systems where a response system y(t) is driven with the output of a driving system x(t), but there is no feedback to the driver. We lose generality but gain tractability with this restriction. To investigate the existence of the synchronization condition y(t)=\ensuremath{\varphi}(x(t)) we introduce the idea of mutual false nearest neighbors to determine when closeness in response space implies closeness in driving space. The synchronization condition also implies that the response dynamics is determined by the drive alone, and we provide tests for this as well. Examples are drawn from computer simulations on various known cases of synchronization and on data from nonlinear electrical circuits. Determining the presence of generalized synchronization will be quite important when one has only scalar observations from the drive and from the response systems since the use of time delay (or other) embedding methods will produce ``imperfect'' coordinates in which strict equality of the synchronized variables is unlikely to transpire.