Spiralling dynamics near heteroclinic networks

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.

change in one figure

Cited by in corpus (6)

• Dense heteroclinic tangencies near a Bykov cycle
• Switching in heteroclinic networks
• Construction of heteroclinic networks in ${\mathbb{R}}^4$
• Torus-breakdown near a Bykov attractor: a case study
• Travelling waves and heteroclinic networks in models of spatially-extended cyclic competition
• Is there switching without suspended horseshoes?