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

Authors

  Henk Broer, Dept. of Mathematics, University of Groningen,
              PO Box 800, 9700 AV Groningen, The Netherlands.
              E-mail: H.W.Broer@math.rug.nl ,
  Carles Sim\'o, Dept. de Matem\`atica Aplicada i An\`alisi, 
              Universitat de Barcelona, Gran Via 585, 08007 Barcelona,
              Spain. E-mail: carles@maia.ub.es and
  Renato Vitolo, Dip. di Matematica ed Informatica, Universit\`a degli
              Studi di Camerino, via Madonna delle Carceri, 
	      62032 Camerino, Italy. E-mail: renato.vitolo@unicam.it

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.