The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: the Arnol'd resonance web

Abstract

A model map $Q$ for the Hopf-saddle-node (HSN) bifurcation of fixed points of diffeomorphisms is studied. The model is constructed to describe the dynamics inside an attracting invariant two-torus which occurs due to the presence of quasi-periodic Hopf bifurcations of an invariant circle, emanating from the central HSN bifurcation. Resonances of the dynamics inside the two-torus attractor yield an intricate structure of gaps in parameter space, the so-called Arnol'd resonance web. Particularly interesting dynamics occurs near the multiple crossings of resonance gaps, where a web of hyperbolic periodic points is expected to occur inside the two-torus attractor. It is conjectured that heteroclinic intersections of the invariant manifolds of the saddle periodic points may give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. To understand this phenomenon, a simple model map of the standard two-torus is developed and studied and the relations with the starting model map $Q$ are discussed.