Title: Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms:
       analysis of a resonance `bubble'

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

  The dynamics near a Hopf-saddle-node bifurcation of fixed points of
  diffeomorphisms is analysed by means of a case study: a two-parameter model
  map $G$ is constructed, such that at the central bifurcation the derivative
  has two complex conjugate eigenvalues of modulus one and one real eigenvalue
  equal to 1. To investigate the effect of resonances, the complex eigenvalues
  are selected to have a 1:5 resonance. It is shown that, near the origin of
  the parameter space, the family $G$ has two secondary Hopf-saddle-node
  bifurcations of period five points.  A cone-like structure exists in the
  neighbourhood, formed by two surfaces of saddle-node and a surface of Hopf
  bifurcations.  Quasi-periodic bifurcations of an invariant circle, forming a
  frayed boundary, are numerically shown to occur in model $G$. Along such
  Cantor-like boundary, an intricate bifurcation structure is detected near a
  1:5 resonance gap.  Subordinate quasi-periodic bifurcations are found
  nearby, suggesting the occurrence of a cascade of quasi-periodic
  bifurcations.