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A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model
Amadeu Delshams, Universitat Politècnica de Catalunya, Barcelona, Spain, Rafael de la Llave, University of Texas, Austin, TX, and Tere M. Seara, Universitat Politècnica de Catalunya, Barcelona, Spain
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Memoirs of the American Mathematical Society
2006; 141 pp; softcover
Volume: 179
ISBN-10: 0-8218-3824-5
ISBN-13: 978-0-8218-3824-2
List Price: US$68
Individual Members: US$40.80
Institutional Members: US$54.40
Order Code: MEMO/179/844
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We introduce a geometric mechanism for diffusion in a priori unstable nearly integrable dynamical systems. It is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. We argue that these objects created by resonances can be incorporated in transition chains taking the place of the destroyed primary KAM tori.

We establish rigorously the existence of this mechanism in a simple model that has been studied before. The main technique is to develop a toolkit to study, in a unified way, tori of different topologies and their invariant manifolds, their intersections as well as shadowing properties of these bi-asymptotic orbits. This toolkit is based on extending and unifying standard techniques. A new tool used here is the scattering map of normally hyperbolic invariant manifolds.

The model considered is a one-parameter family, which for \(\varepsilon = 0\) is an integrable system. We give a small number of explicit conditions the jet of order \(3\) of the family that, if verified imply diffusion. The conditions are just that some explicitely constructed functionals do not vanish identically or have non-degenerate critical points, etc.

An attractive feature of the mechanism is that the transition chains are shorter in the places where the heuristic intuition and numerical experimentation suggests that the diffusion is strongest.

Table of Contents

  • Introduction
  • Heuristic discussion of the mechanism
  • A simple model
  • Statement of rigorous results
  • Notation and definitions, resonances
  • Geometric features of the unperturbed problem
  • Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds
  • The dynamics in \(\tilde \Lambda_{\varepsilon}\)
  • The scattering map
  • Existence of transition chains
  • Orbits shadowing the transition chains and proof of Theorem 4.1
  • Conclusions and remarks
  • An example
  • Acknowledgments
  • Bibliography
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