Boundary crisis in quasiperiodically forced systems

Abstract

We study boundary crisis in quasiperiodically forced dissipative systems using the Hénon map as a characteristic example. The crisis is due to a homoclinic tangency of the stable and unstable manifolds of an accessible invariant circle of saddle type on the basin boundary of the attractor. Numerical evidence shows that the type of boundary crisis can change as the saddle circle loses its accessibility due to the quasiperiodic forcing. This implies the existence of codimension-two double crisis vertices where a curve of boundary crisis and a curve of interior crisis meet. We argue that bifurcation points of higher codimension must exist in the full parameter space.