Periodic forcing of a heteroclinic network

TL;DR

This paper investigates how small periodic perturbations in a specific class of three-dimensional flows can lead to complex dynamics such as strange attractors and horseshoes, starting from a heteroclinic network.

Contribution

It provides a comprehensive mechanism explaining the emergence of complex dynamics in periodically-perturbed heteroclinic networks on three-dimensional manifolds.

Findings

Existence of rotational horseshoes and strange attractors under perturbations

Transition from quasi-periodic tori to chaotic attractors

Explicit example illustrating the theoretical results

Abstract

We present a comprehensive mechanism for the emergence of rotational horseshoes and strange attractors in a class of two-parameter families of periodically-perturbed differential equations defining a flow on a three-dimensional manifold. When both parameters are zero, its flow exhibits an attracting heteroclinic network associated to two periodic solutions. After slightly increasing both parameters, while keeping a two-dimensional connection unaltered, we focus our attention in the case where the two-dimensional invariant manifolds of the periodic solutions do not intersect. We prove a wide range of dynamical behaviour, ranging from an attracting quasi-periodic torus to rotational horseshoes and H\'enon-like strange attractors. We illustrate our results with an explicit example.