 0953 Renato Vitolo, Henk Broer, Carles Simo'
 The Hopfsaddlenode bifurcation for fixed points of
3Ddiffeomorphisms: a computer assisted dynamical inventory
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Mar 23, 09

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Abstract. Dynamical phenomena are studied near a Hopfsaddlenode (HSN) bifurcation of
fixed points of 3Ddiffeomorphisms. 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\'eTakens vector field normal form. The
model dynamics is systematically explored by computation of Lyapunov
exponents and numerical continuation of quasiperiodic invariant circles and
their bifurcations. For an invariant circle, the interaction of 1:5
resonance and quasiperiodic Hopf bifurcations is found to give rise to an
intricate structure of secondary bifurcations of invariant circles and
twotori. This leads to a fractallike pattern of quasiperiodic
bifurcations. Global bifurcations arise in connection with a pair of
saddlefocus fixed points: homoclinic tangencies appear near a spherelike
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 quasiperiodic
period doublings of an invariant circle, followed by loss of reducibility.
The other route involves intermittency due to a quasiperiodic saddlenode
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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