Robust heteroclinic cycles in delay differential equations

Abstract

We consider the existence of heteroclinic cycles in Γ-equivariant delay-differential equations which emerge from symmetry-breaking bifurcations from an equilibrium solution with maximal isotropy subgroup. We begin by describing the existence of robust heteroclinic cycles on finite-dimensional centre manifolds and show that these are also robust to Γ-equivariant perturbations of the delay-differential equation. We then present the first example of a delay-differential equation which supports a heteroclinic cycle not contained within a finite-dimensional submanifold. This system is a delayed version of the Guckenheimer and Holmes equivariant three-dimensional example realized as a coupled cell system. We prove the existence of the heteroclinic cycle and show that it is structurally stable to Γ-equivariant perturbations which preserve certain codimension one subspaces of phase space associated with fixed point subspaces. By letting the cell dynamics be delay-dependent, we show that for a large enough delay, we obtain a heteroclinic cycle joining periodic solutions. Numerical simulations are presented and discussed.