Desynchronization of chaos in coupled logistic maps

When identical chaotic oscillators interact, a state of complete or partial synchronization may be attained in which the motion is restricted to an in variant manifold of lower dimension than the full phase space. Riddling of the basin of attraction arises when particular orbits embedded in the synch ronized chaotic state become transversely unstable while the state remains attracting on the average. Considering a system of two coupled logistic map s, we show that the transition to riddling will be soft or hard, depending on whether the first orbit to lose its transverse stability undergoes a sup ercritical or subcritical bifurcation. A subcritical bifurcation can lead d irectly to global riddling of the basin of attraction for the synchronized chaotic state. A supercritical bifurcation, on the other hand, is associate d with the formation of a so-called mixed absorbing area that stretches alo ng the synchronized chaotic state, and from which trajectories cannot escap e. This gives rise to locally riddled basins of attraction. We present thre e different scenarios for the onset of riddling and for the subsequent tran sformations of the basins of attraction. Each scenario is described by foll owing the type and location of the relevant asynchronous cycles, and determ ining their stable and unstable invariant manifolds. One scenario involves a contact bifurcation between the boundary of the basin of attraction and t he absorbing area. Another scenario involves a long and interesting series of bifurcations starting with the stabilization of the asynchronous cycle p roduced in the riddling bifurcation and ending in a boundary crisis where t he stability of an asynchronous chaotic state is destroyed. Finally, a phas e diagram is presented to illustrate the parameter values at which the vari ous transitions occur. [S1063-651X(99)04509-2].