Switching near a network of rotating nodes

Abstract

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, 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.

Global generic dynamics close to symmetry

“…Item 3 is a direct consequence of the main result about finite and infinite switching of Aguiar et al [5].…”

“…The main result of [5] is that, close to what remains of the network Σ after perturbation, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections in any given order. This is the concept of heteroclinic switching; the next paragraph gives a set-up of switching near a heteroclinic network.…”

“…The next lemma summarizes some basic technical results about the geometry near the saddle-foci. The proof may be found in Section 6 of Aguiar et al [5] -this is why it will be omitted here. …”

“…In a more general setting the dynamics around heteroclinic cycles has been studied by Aguiar et al [5]. The main result of [5] is that, close to what remains of the network Σ after perturbation, there are trajectories that visit neighbourhoods of the saddles following all the heteroclinic connections in any given order.…”

“…On the proof of item 3 of Lemma 10, the authors of [5] used implicitly that the orientations in which trajectories turn around in V and W are the same (i.e., they used implicitly property (P8)).…”