Switching near a network of rotating nodes

Abstract

We study the dynamics of a Z 2 ⊕ Z 2-equivariant vector field in the neighbourhood of a heteroclinic network with a periodic trajectory and symmetric equilibria. We assume that around each equilibrium the linearization of the vector field has non-real eigenvalues. Trajectories starting near each node of the network turn around in space either following the periodic trajectory or due to the complex eigenvalues near the equilibria. Thus, in a network with rotating nodes, the rotations combine with transverse intersections of two-dimensional invariant manifolds to create switching near the network; close to the network, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections of the network in any given order. Our results are motivated by an example where switching was observed numerically by forced symmetry breaking of an asymptotically stable network with O(2) symmetry.