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.