Asymptotics in a family of linked strip maps

We study the asymptotic regime of the limiting integrable system analytically, and numerically some features of its very rich near-integrable dynamics. We unveil a dichotomy: there is one regime in which the nonlinearity tends to zero, and a second where it does not. In the latter case, numerical experiments suggest that the distribution of the periods of orbits is consistent with that of random dynamics; in the former case the fluctuations result in an intricate structure of resonances.