Diffusion and Drift in Volume-Preserving Maps

A nearly-integrable dynamical system has a natural formulation in terms of actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with frequency $\mathrm{\Omega }(y)$). We study angle-action maps that are close to symplectic and have a positive-definite twist, the derivative of the frequency map, $D\mathrm{\Omega }(y)$. 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.