Abundance of Strange Attractors Near an Attracting Periodically Perturbed Network

Abstract

We study the dynamics of the periodically forced May--Leonard system. We extend previous results on the field, and we identify different dynamical regimes depending on the strength of attraction $\delta$ of the network and the frequency $\omega$ of the periodic forcing. We focus our attention in the case $\delta\gg1$ and $\omega \approx 0$, where we show that, for a positive Lebesgue measure set of parameters (amplitude of the periodic forcing), the dynamics are dominated by strange attractors with fully stochastic properties, supporting Sinai--Ruelle--Bowen (SRB) measures. The proof is performed by using the Wang and Young's theory of rank-one strange attractors. This work ends the discussion about the existence of observable and sustainable chaos in this scenario. We also identify some bifurcations occurring in the transition from an attracting two-torus to rank-one strange attractors, whose existence has been suggested by numerical simulations.