A minimal system with a depth-two heteroclinic network

Abstract

We present an example of a three-dimensional ordinary differential equation with an invariant set that is a heteroclinic network of depth two (a heteroclinic network with a hierarchical structure) between hyperbolic equilibria. This is the first example of minimal dimension (namely, three dimensions) that is robust for a constrained system of ODEs – the constraint is that the flow preserves the boundaries of an invariant cube and an axis through opposite faces of the cube. We examine the system from both analytical and numerical viewpoints, and in a neighbourhood of the network we derive a reduced, one-dimensional, return map where we observe a variety of complicated dynamical behaviours. This includes co-existence of infinitely many stable limit cycles, and we present evidence that there may be infinitely many chaotic attractors within this neighbourhood.

Keywords:

• heteroclinic network
• attractor