Symmetry Breaking Bifurcations of Chaotic Attractors

In an array of coupled oscillators synchronous chaos may occur in the sense that all the oscillators behave identically although the corresponding motion is chaotic. When a parameter is varied this fully symmetric dynamical state can lose its stability, and the main purpose of this paper is to investigate which type of dynamical behavior is expected to be observed once the loss of stability has occurred. The essential tool is a classiication of Lyapunov exponents based on the symmetry of the underlying problem. This classiication is crucial in the derivation of the analytical results but it also allows an eecient computation of the dominant Lyapunov exponent associated with each symmetry type. We show how these dominant exponents determine the stability of invariant sets possessing various instantaneous symmetries and this leads to the idea of symmetry breaking bifurcations of chaotic attractors. Finally the results and ideas are illustrated for several systems of coupled oscillators.