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.