"Large'' strange attractors in the unfolding of a Hopf-zero singularity

Alexandre Rodrigues, ISEG (Instituto Superior de Economia e Gestão) da Universidade de Lisboa

Abstract:

In this talk, we derive theoretical results to show the occurrence of strange attractors in some analytic unfoldings of a Hopf-zero singularity.

More precisely, we present a comprehensive mechanism for the emergence of strange attractors in a two-parametric family of differential equations acting on a threedimensional sphere. When both parameters are zero, its flow exhibits an attracting heteroclinic network made by two 1-dimensional connections and one 2-dimensional separatrix between two hyperbolic saddles-foci with different Morse indices. After slightly increasing both parameters, while keeping the one-dimensional connections unaltered, we focus our attention on the case where the two-dimensional invariant manifolds of the equilibria do not intersect. We prove the existence of many complicated dynamical objects, ranging from an attracting quasi-periodic torus to Hénon-like strange attractors, as a consequence of the Torus-Breakdown Theory. 

The mechanism for the creation of horseshoes and strange attractors is discussed.