Role of invariant and minimal absorbing areas in chaos synchronization

Abstract

In this paper the method of critical curves, a tool for the study of the global dynamical properties of two-dimensional noninvertible maps, is applied to the study of chaos synchronization and related phenomena of riddling, blowout, and on-off intermittency. A general procedure is suggested in order to obtain the boundary of a particular two-dimensional compact trapping region, called absorbing area, containing the one-dimensional chaotic set on which synchronized dynamics occur. The main purpose of the paper is to show that only invariant and minimal absorbing areas are useful to characterize the global dynamical behavior of the dynamical system when a Milnor attractor with locally riddled basin or a chaotic saddle exists, and may strongly influence the effects of riddling and blowout bifurcations. Some examples are given for a system of two coupled logistic maps, and some practical methods and numerical tricks are suggested in order to ascertain the properties of invariance and minimality of an absorbing area. Some general considerations are given concerning the transition from locally riddled to globally riddled basins, and the role of the absorbing area in the occurrence of such transition is discussed.