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)
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.