Transverse instability and riddled basins in a system of two coupled logistic maps

Riddled basins denote a characteristic type of fractal domain of attraction that can arise when a chaotic motion is restricted to an invariant subspace of total phase space. An example is the synchronized motion of two identical chaotic oscillators. The paper examines the conditions for the appearance of such basins for a system of two symmetrically coupled logistic maps. We determine the regions in parameter plane where the transverse Lyapunov exponent is negative. The bifurcation curves for the transverse destabilization of low-periodic orbits embedded in the chaotic attractor are obtained, and we follow the changes in the attractor and its basin of attraction when scanning across the riddling and blowout bifurcations. It is shown that the appearance of transversely unstable orbits does not necessarily lead to an observable basin 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 may break, and the attractor may spread into two-dimensional phase space.