Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: analysis of a resonance ‘bubble’

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. Dept. of Mathematics, University of Groningen, PO Box 800, 9700 AV Groningen, The Netherlands. E-mail: [email protected] Dept. de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007 Barcelona, Spain. E-mail: [email protected] Corresponding author address: Dr. Renato Vitolo, Dip. di Matematica ed Informatica, Università degli Studi di Camerino, via Madonna delle Carceri, 62032 Camerino, Italy. E-mail: [email protected]