Spiralling dynamics near heteroclinic networks

TL;DR

This paper explicitly constructs a family of vector fields on the 3-sphere exhibiting complex, chaotic dynamics near heteroclinic networks, including horseshoes and Newhouse phenomena, with analytical proofs.

Contribution

It provides explicit examples of vector fields with analytically provable complex dynamics involving heteroclinic networks and chaotic invariant sets.

Findings

Existence of a spiralling attractor with heteroclinic connections

Presence of topological horseshoes with increasing complexity

Identification of bifurcations leading to chaos

Abstract

There are few explicit examples in the literature of vector fields exhibiting complex dynamics that may be proved analytically. We construct explicitly a {two parameter family of vector fields} on the three-dimensional sphere $\EU^3$, whose flow has a spiralling attractor containing the following: two hyperbolic equilibria, heteroclinic trajectories connecting them {transversely} and a non-trivial hyperbolic, invariant and transitive set. The spiralling set unfolds a heteroclinic network between two symmetric saddle-foci and contains a sequence of topological horseshoes semiconjugate to full shifts over an alphabet with more and more symbols, {coexisting with Newhouse phenonema}. The vector field is the restriction to $\EU^3$ of a polynomial vector field in $\RR^4$. In this article, we also identify global bifurcations that induce chaotic dynamics of different types.