TRANSVERSE INSTABILITY AND RIDDLED BASINS IN A SYSTEM OF 2 COUPLED LOGISTIC MAPS

Riddled basins denote a characteristic type of fractal domain of attra ction that can arise when a chaotic motion is restricted to an invaria nt subspace of total phase space. An example is the synchronized motio n of two identical chaotic oscillators. The paper examines the conditi ons for the appearance of such basins for a system of two symmetricall y coupled logistic maps. We determine the regions in parameter plane w here the transverse Lyapunov exponent is negative. The bifurcation cur ves for the transverse destabilization of low-periodic orbits embedded in the chaotic attractor are obtained, and we follow the changes in t he attractor and its basin of attraction when scanning across the ridd ling and blowout bifurcations. It is shown that the appearance of tran sversely unstable orbits does not necessarily lead to an observable ba sin riddling, and that the loss of weak stability (when the transverse Lyapunov exponent becomes positive) does not necessarily destroy the basin of attraction. Instead, the symmetry of the synchronized state m ay break, and the attractor may spread into two-dimensional phase spac e.