Diffusion and drift in volume-preserving maps

Abstract

A nearly-integrable dynamical system has a natural formulation in terms of actions, y (nearly constant), and angles, x (nearly rigidly rotating with frequency Ω(y)).We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, DΩ(y), that is positive-definite. When the map is symplectic, Nekhoroshev’s theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-r resonances. A comparison with computations for a generalized Froeschl´e map in four-dimensions shows that this theory gives accurate results for the rank-one case.