Arnold diffusion in weakly coupled standard maps.

Abstract

The standard map has a divided phase space in which two-dimensional regions of stochasticity are isolated by one-dimensional Kolmogorov-Arnold-Moser (KAM) curves that form a barrier to diffusion in action. When two standard maps are coupled together, the two-dimensional KAM surfaces no longer divide the four-dimensional phase space, and particles diffuse slowly along stochastic layers by the process of Arnold diffusion. We compare an analytic calculation of the rate of localized Arnold diffusion with numerically determined rates in regions having rotational and librational KAM curves for a single map, in the weak-coupling limit for which the three-resonance model holds. We then determine the rate of global Arnold diffusion across many cells of the 2\ensuremath{\pi}-periodic mapping. The global diffusion rate depends both on the local diffusion rate and on the relative volume occupied by the various stochastically accessible regions in the four-dimensional phase space.