Henk Broer, Carles Simo, Renato Vitolo
Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: 
the Arnold resonance web
(15232K, PS)

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.