Resonances in skew and reducible quasi-periodic Hopf bifurcations

Abstract

Two types of quasi-periodic Hopf bifurcations are known, in which a Whitney smooth family of quasi-periodic attractors loses its hyperbolicity. One is the reducible case, where the normal linear dynamics are trivial. Another is the skew case, where the normal dynamics are topologically non-trivial. There, the dynamics can involve periodicity, quasi-periodicity and chaos, including chaos with a mixed spectrum. This paper investigates a model system where the bifurcating circle supports Morse-Smale (also called resonant) dynamics. First this is done under the assumption of rotational symmetry (in the normal direction) of the system; a local bifurcation analysis is given for this. Then the effects of generic (non-symmetric) perturbations are discussed. It turns out that the bifurcation diagram is organized by several codimension-two bifurcations: saddle-Hopf and degenerate Hopf bifurcations for diffeomorphisms, and Bogdanov-Takens bifurcations of invariant circles. The (local) bifurcation diagram is computed analytically for the model system. We believe that this model is representative for the minimal dynamical complexity which generically occurs whenever the dynamics on the bifurcating circle is resonant.