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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
Tere M. Seara Universitat Politècnica de Catalunya, Barcelona, Spain
Available Formats:
Electronic ISBN: 978-1-4704-0445-1
Product Code: MEMO/179/844.E
List Price: $72.00 MAA Member Price:$64.80
AMS Member Price: $43.20 Click above image for expanded view 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 Tere M. Seara Universitat Politècnica de Catalunya, Barcelona, Spain Available Formats:  Electronic ISBN: 978-1-4704-0445-1 Product Code: MEMO/179/844.E  List Price:$72.00 MAA Member Price: $64.80 AMS Member Price:$43.20
• Book Details

Memoirs of the American Mathematical Society
Volume: 1792006; 141 pp
MSC: Primary 37; 34; Secondary 70;

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.

• Chapters
• 1. Introduction
• 2. Heuristic discussion of the mechanism
• 3. A simple model
• 4. Statement of rigorous results
• 5. Notation and definitions, resonances
• 6. Geometric features of the unperturbed problem
• 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds
• 8. The dynamics in $\tilde {\Lambda }_\epsilon$
• 9. The scattering map
• 10. Existence of transition chains
• 11. Orbits shadowing the transition chains and proof of Theorem 4.1
• 12. Conclusions and remarks
• 13. An example
• Requests

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Volume: 1792006; 141 pp
MSC: Primary 37; 34; Secondary 70;

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.

• Chapters
• 1. Introduction
• 2. Heuristic discussion of the mechanism
• 3. A simple model
• 4. Statement of rigorous results
• 5. Notation and definitions, resonances
• 6. Geometric features of the unperturbed problem
• 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds
• 8. The dynamics in $\tilde {\Lambda }_\epsilon$
• 9. The scattering map
• 10. Existence of transition chains
• 11. Orbits shadowing the transition chains and proof of Theorem 4.1
• 12. Conclusions and remarks
• 13. An example
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