 07187 Henk Broer, Carles Simo, Renato Vitolo
 Hopfsaddlenode bifurcation for fixed points of 3Ddiffeomorphisms: analysis of a resonance `bubble'
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Aug 2, 07

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Abstract. The dynamics near a Hopfsaddlenode bifurcation of fixed points of diffeomorphisms is analysed by means of a case study: a twoparameter 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 Hopfsaddlenode bifurcations of period five points. A conelike structure exists in the neighbourhood, formed by two surfaces of saddlenode and a surface of Hopf bifurcations. Quasiperiodic bifurcations of an invariant circle, forming a frayed boundary, are numerically shown to occur in model G. Along such Cantorlike boundary, an intricate bifurcation structure is detected near a 1:5 resonance gap. Subordinate quasiperiodic bifurcations are found nearby, suggesting the occurrence of a cascade of quasiperiodic bifurcations.
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