Effect of noise on residence times of a heteroclinic cycle

Abstract

A heteroclinic cycle is an invariant set in a dynamical system consisting of saddle-type equilibria and heteroclinic connections between them. It is known that deterministic perturbations (inputs) to a heteroclinic cycle generally lead to periodic solutions. Addition of noise to such a system leads to a non-intuitive result: there is a range of noise levels for which the mean residence time near the equilibria of the heteroclinic cycle increases as the noise level increases to a given threshold. We explain how the interaction between noise and inputs gives rise to this by combining analytical results from constructing a Poincaré map with a simple stochastic system. We support our results with numerical simulations.

Keywords:

• Heteroclinic cycle
• noise
• residence time
• periodic solution

Acknowledgments

The authors would like to thank P. Ashwin for valuable discussions and an anonymous referee for helpful comments on an earlier version.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by the Marsden Fund Council from New Zealand Government funding, managed by The Royal Society Te Apārangi [Grant No. 17-UOA-096].