Routes to chaos in the Hopf-saddle-node bifurcation for fixed points of 3D-diffeomorphisms

Dynamical phenomena are studied near a Hopf-saddle-node bifurcation of fixed points of 3D-diffeomorphisms. The interest lies in the neighbourhood of weak resonances of the complex conjugate eigenvalues. The 1 : 5 case is chosen here because it has the lowest order among the weak resonances, and therefore it is likely to have a most visible influence on the bifurcation diagram. A model map is obtained by a natural construction, through perturbation of the flow of a Poincaré–Takens normal form vector field. Global bifurcations arise in connection with a pair of saddle-focus fixed points: homoclinic tangencies occur 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 three routes are described. One route involves a sequence of quasi-periodic period doublings of an invariant circle where loss of reducibility also takes place during the process. A second route involves intermittency due to a quasi-periodic saddle-node bifurcation of an invariant circle. Finally a route involving heteroclinic phenomena is discussed. Multistability occurs in several parameter subdomains: we analyse the structure of the basins for a case of coexistence of a strange and a quasi-periodic attractor and for coexistence of two strange attractors. By construction, the phenomenology of the model map is expected in generic families of 3D diffeomorphisms.