Papers International Journal of Bifurcation and Chaos, Vol. 10, No. 10 (2000) 2323–2337 c  World Scientific Publishing Company FRACTAL DIMENSION FOR POINCAR ´ E RECURRENCES AS AN INDICATOR OF SYNCHRONIZED CHAOTIC REGIMES VALENTIN S. AFRAIMOVICH Instituto de Investigacion en Comunicaci´ on Optica, Universidad Autonoma de San Luis Potosi, San Luis Potosi 78000, Mexico WEN-WEI LIN Department of Mathematics, National Tsing-Hua University, Hsinchu, Taiwan R.O.C. NIKOLAI F. RULKOV Institute for Nonlinear Science, University of California, San Diego, La Jolla, CA 92093-0402, USA Received August 24, 1999; Revised November 1, 1999 The studies of the phenomenon of chaos synchronization are usually based upon the analysis of the existence of transversely stable invariant manifold that contains an invariant set of tra- jectories corresponding to synchronous motions. In this paper we develop a new approach that relies on the notions of topological synchronization and the dimension for Poincar´ e recurrences. We show that the dimension of Poincar´ e recurrences may serve as an indicator for the onset of synchronized chaotic oscillations. This indicator is capable of detecting the regimes of chaos synchronization characterized by the frequency ratio p : q . 1. Introduction It is well-known that coupling between the dissipa- tive dynamical systems with chaotic behavior can result in the onset of synchronized chaotic oscilla- tions (see e.g. [Pecora & Carroll, 1998] and refer- ences therein). In other words, a system ˙ x = f (x)+ cF (x, y, c) , ˙ y = g(y)+ cG(x, y, c) , (1) where x ∈ ℜ m ,y ∈ ℜ n , and c is the cou- pling parameter, can behave in such a way that the x-component and y-component of solution [x(t, x 0 ,y 0 ),y(t, x 0 ,y 0 )] manifest some type of synchrony for t ≥ t 0 ≫ 1, independent of initial conditions (x 0 ,y 0 ) in a large region of ℜ n+m . The most simple type of synchronous chaotic behavior is the regime of identical synchronization. In this regime the solutions of the coupled oscilla- tors (1) satisfy the following property lim t→∞ |x(t, x 0 ,y 0 ) − y(t, x 0 ,y 0 )| =0 . (2) Of course, in order to achieve this type of behavior in the case m = n, the right-hand side of the system (1) should satisfy the identity f (x)+ cF (x, x, c)= g(x)+ cG(x, x, c) . (3) For example, it is so if f (x) ≡ g(x) and F (x, x, c)= G(x, x, c) ≡ 0. It is easy to see, that when the iden- tity (3) holds the system (1) has a manifold x = y, 2323