Continued Fractions Hierarchy of Rotation Numbers in Planar Dynamics

Abstract

Global bifurcations such as crises of attractors, explosions of chaotic saddles, and metamorphoses of basin boundaries play a crucial role in understanding the dynamical evolution of physical systems. Global bifurcations in dissipative planar maps are typically caused by collisions of invariant manifolds of periodic orbits, whose dynamical behaviors are described by rotation numbers. We show that the rotation numbers of the periodic orbits created at certain important tangencies are determined by the continued fraction expansion of the rotation number of the orbit involved in the collision.