  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 a Sinai-Ruelle-Bowen (SRB) measure. The proof is performed by using
the Wang and Young 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.
