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.
    Keywords: symplectic maps, Nekhoroshev’s theorem, chaotic transport
    Citation: Guillery N., Meiss J. D., Diffusion and Drift in Volume-Preserving Maps, Regular and Chaotic Dynamics, 2017, Volume 22, Number 6, pp. 700–720

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