Bifurcations from an attracting heteroclinic cycle under periodic forcing

TL;DR

This paper analyzes how periodic forcing affects a heteroclinic cycle on the two-sphere, revealing bifurcations, invariant curves, and complex dynamics through a reduced two-dimensional map.

Contribution

It derives the first return map for a perturbed heteroclinic network and uncovers bifurcation phenomena and complex dynamics in a non-autonomous setting.

Findings

Existence of an attracting invariant closed curve for small perturbations

Bistability near frequency locking points with coexistence of invariant curve and fixed point

Bifurcation into periodic solutions and chaotic dynamics as perturbation increases

Abstract

There are few examples of non-autonomous vector fields exhibiting complex dynamics that may be proven analytically. We analyse a family of periodic perturbations of a weakly attracting robust heteroclinic network defined on the two-sphere. We derive the first return map near the heteroclinic cycle for small amplitude of the perturbing term, and we reduce the analysis of the non-autonomous system to that of a two-dimensional map on a cylinder. Interesting dynamical features arise from a discrete-time Bogdanov-Takens bifurcation. When the perturbation strength is small the first return map has an attracting invariant closed curve that is not contractible on the cylinder. Near the centre of frequency locking there are parameter values with bistability: the invariant curve coexists with an attracting fixed point. Increasing the perturbation strength there are periodic solutions that bifurcate into a closed contractible invariant curve and into a region where the dynamics is conjugate to a full shift on two symbols.