Three-dimensional horseshoes near an unfolding of a Hopf-Hopf singularity

Abstract

Motivated by a certain type of unfolding of a Hopf-Hopf singularity, we consider a one-parameter family ${\left(f_{\gamma }\right)}_{\gamma \ge 0}$ of $C^3$–vector fields in $R^4$ whose flows exhibit a heteroclinic cycle associated to two periodic solutions and a bifocus, all of them hyperbolic. It is formally proved that combining rotation with a generic condition concerning the transverse intersection between the three-dimensional invariant manifolds of the periodic solutions, all sets are highly distorted by the first return map and hyperbolic three-dimensional horseshoes emerge, accumulating on the network. Infinitely many linked horseshoes prompt the coexistence of infinitely many saddle-type invariant sets for all values of $\gamma \gtrsim 0$ belonging to the heteroclinic class of the two hyperbolic periodic solutions. We apply the results to a particular unfolding of the Hopf-Hopf singularity, the so-called Gaspard-type unfolding.