09-53 Renato Vitolo, Henk Broer, Carles Simo'

The Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms: a computer assisted dynamical inventory (12712K, gzipped Postscript) Mar 23, 09

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Abstract. Dynamical phenomena are studied near a Hopf-saddle-node (HSN) bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues and the $1:5$ case is considered here. A model map is obtained by a natural construction, through perturbation of the Poincar\'e-Takens vector field normal form. The model dynamics is systematically explored by computation of Lyapunov exponents and numerical continuation of quasi-periodic invariant circles and their bifurcations. For an invariant circle, the interaction of 1:5 resonance and quasi-periodic Hopf bifurcations is found to give rise to an intricate structure of secondary bifurcations of invariant circles and two-tori. This leads to a fractal-like pattern of quasi-periodic bifurcations. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies appear near a sphere-like heteroclinic structure formed by the 2D stable and unstable manifolds of the saddle points. Strange attractors occur for nearby parameter values and two routes are described. One route involves a finite number of quasi-periodic period doublings of an invariant circle, followed by loss of reducibility. The other route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.

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