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.

Keywords:

• delay-differential equations
• equivariance
• heteroclinic cycles
• symmetry-breaking bifurcations
• realization theorems

AMS Subject Classifications::

• 37G40
• 34K18
• 37C29

Acknowledgements

I would like to thank A. Palacios for interesting discussions on this topic and in particular for pointing me to the thesis of Longhini. This research is supported by the Natural Sciences and Engineering Research Council of Canada in the form of a Discovery Grant and by UOIT.