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 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 positi...

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Bibliographic Details
Published in:Regular & chaotic dynamics 2017-11, Vol.22 (6), p.700-720
Main Authors: Guillery, Nathan, Meiss, James D.
Format: Article
Language:English
Subjects:
Drift
Dynamical Systems and Ergodic Theory
Mathematics
Mathematics and Statistics
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Summary: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.
ISSN:1560-3547
1560-3547
1468-4845
DOI:10.1134/S1560354717060089