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Vizualizing the Dynamics of Symplectic Maps

Introductory Workshop: Hamiltonian systems, from topology to applications through analysis August 20, 2018 - August 24, 2018

August 24, 2018 (11:00 AM PDT - 12:00 PM PDT)
Speaker(s): James Meiss (University of Colorado at Boulder)
Location: MSRI: Simons Auditorium
  • symplectic maps

  • transport

  • resonance

  • chaotic dynamics

Primary Mathematics Subject Classification
Secondary Mathematics Subject Classification No Secondary AMS MSC


A nearly-integrable dynamical system has a natural formulation in terms of actions that are nearly constant and angles that nearly rigidly rotate. Such maps arise naturally in the Hamiltonian, or symplectic case where the variables appear in canonical pairs. We study angle-action maps that are close to symplectic and have a twist 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 Froeschlé map in four-dimensions, shows that this theory gives accurate results for the rank-one case.

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Video/Audio Files


H.264 Video 16-Meiss.mp4 196 MB video/mp4 Download
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