Interior crises in quasiperiodically forced period-doubling systems

As a representative model for quasiperiodically forced period-doubling systems, we consider the quasiperiodically forced logistic map, and investigate the dynamical mechanism for the interior crises. For small quasiperiodic forcing ε, a chaotic attractor abruptly widens via a “standard” interior crisis when it collides with a smooth unstable torus. However, as ε passes a threshold value, the smooth unstable torus loses its accessibility from the interior of the basin of the attractor. For this case, we use the rational approximation to the quasiperiodic forcing, and find that a nonstandard interior crisis occurs for a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor when it collides with an invariant “ring-shaped” unstable set. Particularly, we note that a three-band smooth torus transforms into a single-band intermittent strange nonchaotic attractor through the nonstandard interior crisis. The intermittent strange nonchaotic attractor is also characterized in terms of the average interburst time and the local Lyapunov exponent.