A Geometric Mechanism For Diffusion In Hamiltonian Systems Overcoming The Large Gap Problem Heuristics And Rigorous Verification On A Model
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A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model
Author: Amadeu Delshams
language: en
Publisher: American Mathematical Soc.
Release Date: 2006
Beginning by introducing a geometric mechanism for diffusion in a priori unstable nearly integrable dynamical systems. This book is based on the observation that resonances, besides destroying the primary KAM tori, create secondary tori and tori of lower dimension. It argues that these objects created by resonances can be incorporated in transition chains taking the place of the destroyed primary KAM tori.The authors establish rigorously the existence of this mechanism in a simplemodel 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 unifyingstandard 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 themechanism is that the transition chains are shorter in the places where the heuristic intuition and numerical experimentation suggests that the diffusion is strongest.
Hamiltonian Dynamical Systems and Applications
Author: Walter Craig
language: en
Publisher: Springer Science & Business Media
Release Date: 2008-02-17
This volume is the collected and extended notes from the lectures on Hamiltonian dynamical systems and their applications that were given at the NATO Advanced Study Institute in Montreal in 2007. Many aspects of the modern theory of the subject were covered at this event, including low dimensional problems. Applications are also presented to several important areas of research, including problems in classical mechanics, continuum mechanics, and partial differential equations.
Hamiltonian Systems
Author: Albert Fathi
language: en
Publisher: Cambridge University Press
Release Date: 2024-05-31
A selection of results, spanning a broad spectrum of disciplines, from the MSRI program on Hamiltonian Systems during Fall 2018.