A new mechanism for stability loss from a heteroclinic cycle

Abstract

Asymptotically stable robust heteroclinic cycles can lose stability through resonance or transverse bifurcations. In a transverse bifurcation, an equilibrium in the cycle undergoes a local bifurcation, causing a change in stability. A resonance bifurcation is a global phenomenon, determined by an algebraic condition on the eigenvalues, and is generically accompanied by the birth or death of a long-period periodic orbit. In this article we demonstrate a new mechanism causing loss of stability, which is neither resonant nor transverse in the usual sense. The location of the instability is determined by an algebraic condition on the eigenvalues, but the instability occurs in a transverse direction. Furthermore, after the bifurcation, when the cycle is unstable, open sets of trajectories are seen to initially approach the network for an extended period, before moving away in the unstable direction. This should serve as a warning to all those doing numerics near heteroclinic cycles who deduce that the cycle is stable merely because trajectories are observed to initially approach the cycle.

Keywords:

• robust heteroclinic cycles
• bifurcations with symmetry

Acknowledgements

This work came about after a search for a ‘small’ heteroclinic network with switching behaviour, inspired by discussions with Alastair Rucklidge and Vivien Kirk. The author is very grateful to two anonymous referees for their careful reading of the manuscript and helpful comments on an earlier draft.