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Regular and Chaotic Dynamics

, Volume 22, Issue 1, pp 78–108 | Cite as

Arnold diffusion for a complete family of perturbations

  • Amadeu DelshamsEmail author
  • Rodrigo G. Schaefer
On the 70th Birthday of Nikolai N. Nekhoroshev Special Memorial Issue. Part 2

Abstract

In this work we illustrate the Arnold diffusion in a concrete example — the a priori unstable Hamiltonian system of 2 + 1/2 degrees of freedom H(p, q, I, φ, s) = p 2/2+ cos q − 1 + I 2/2 + h(q, φ, s; ε) — proving that for any small periodic perturbation of the form h(q, φ, s; ε) = ε cos q (a 00 + a 10 cosφ + a 01 cos s) (a 10 a 01 ≠ 0) there is global instability for the action. For the proof we apply a geometrical mechanism based on the so-called scattering map. This work has the following structure: In the first stage, for a more restricted case (I* ~ π/2μ, μ = a 10/a 01), we use only one scattering map, with a special property: the existence of simple paths of diffusion called highways. Later, in the general case we combine a scattering map with the inner map (inner dynamics) to prove the more general result (the existence of instability for any μ). The bifurcations of the scattering map are also studied as a function of μ. Finally, we give an estimate for the time of diffusion, and we show that this time is primarily the time spent under the scattering map.

Keywords

Arnold diffusion normally hyperbolic invariant manifolds scattering maps 

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Copyright information

© Pleiades Publishing, Ltd. 2017

Authors and Affiliations

  1. 1.Department de MatemàtiquesUniversitat Politècnica de CatalunyaBarcelonaSpain

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