The Hopf-saddle-node bifurcation for fixedpoints of 3D-diffeomorphisms: the Arnol'dresonance web

摘要

A model map Q for the Hopf-saddle-node (HSN) bifurcation of fixed pointsof diffeomorphisms is studied. The model is constructed to describe the dy-namics inside an attracting invariant two-torus which occurs due to the pres-ence of quasi-periodic Hopf bifurcations of an invariant circle, emanating fromthe central HSN bifurcation. Resonances of the dynamics inside the two-torusattractor yield an intricate structure of gaps in parameter space, the so-calledArnol'd resonance web. Particularly interesting dynamics occurs near the mul-tiple crossings of resonance gaps, where a web of hyperbolic periodic points isexpected to occur inside the two-torus attractor. It is conjectured that hete-roclinic intersections of the invariant manifolds of the saddle periodic pointsmay give rise to the occurrence of strange attractors contained in the two-torus. This is a concrete route to the Newhouse-Ruelle-Takens scenario. Tounderstand this phenomenon, a simple model map of the standard two-torusis developed and studied and the relations with the starting model map Q arediscussed.