Mechanism for boundary crises in quasiperiodically forced period-doubling systems

We investigate the mechanism for boundary crises in the quasiperiodically forced logistic map which is a representative model for quasiperiodically forced period-doubling systems. For small quasiperiodic forcing ɛ, a chaotic attractor disappears suddenly via a “standard” boundary crisis when it collides with the smooth unstable torus. However, when passing a threshold value of ɛ, a basin boundary metamorphosis occurs, and then the smooth unstable torus is no longer accessible from the interior of the basin of the attractor. For this case, using the rational approximations to the quasiperiodic forcing, it is shown that a nonchaotic attractor (smooth torus or strange nonchaotic attractor) as well as a chaotic attractor is destroyed abruptly through a new type of boundary crisis when it collides with an invariant “ring-shaped” unstable set which has no counterpart in the unforced case.