# Diffusion and Drift in Volume-Preserving Maps

*2017, Volume 22, Number 6, pp. 700–720*

Author(s):

**Guillery N., Meiss J. D.**

A nearly-integrable dynamical system has a natural formulation in terms of actions, $y$ (nearly constant), and angles, $x$ (nearly rigidly rotating with frequency $\Omega(y)$). We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, $D\Omega(y)$, that is positive-definite. When the map is symplectic, NekhoroshevЃfs 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 Froeschlé map in four-dimensions shows that this theory gives accurate results for the rank-one case.

Access to the full text on the Springer website |