A Geometric Mechanism for Diffusion in Hamilto. Heuristics and Rigorous Verification on a Model. [Memoirs of the American Mathematical Society, Vol. 844].

Delshams, Amadeu, Rafael De LA Llave and Tere M. Seara:

ISBN 10: 0821838245 / ISBN 13: 9780821838242

Published by American Mathematical Society, 2006

About this Item

141 p. Very good. Shrink wrapped. / Sehr guter Zustand. In Folie verschweißt. Sprache: Englisch Gewicht in Gramm: 343.

Seller Inventory # 824894

Bibliographic Details

• Title: A Geometric Mechanism for Diffusion in Hamilto. Heuristics and Rigorous Verification on a Model. [Memoirs of the American Mathematical Society, Vol. 844].
• Publisher: American Mathematical Society
• Publication Date: 2006
• Language: English
• ISBN 10: 0821838245
• ISBN 13: 9780821838242
• Binding: Softcover/Paperback
• Condition: Sehr gut
• Weight: 343 grams
• Seller catalogs:
  • Mathematik

About this title

Synopsis

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.