TY  - BOOK
AU  - Delshams,Amadeu
AU  - de la Llave,Rafael
AU  - Seara,Tere M.
TI  - Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model
T2  - Memoirs of the American Mathematical Society
SN  - 9781470404451
AV  - QA614.833 -- .D45 2006eb 
U1  - 510 s;515/.39 
PY  - 2005///
CY  - Providence
PB  - American Mathematical Society
KW  - Nonholonomic dynamical systems
KW  - Mechanics
KW  - Differential equations -- Qualitative theory
KW  - Electronic books
N1  - Intro -- Contents -- Chapter 1. Introduction -- Chapter 2. Heuristic discussion of the mechanism -- 2.1. Integrable systems, resonances, secondary tori -- 2.2. Heuristic description of the mechanism -- Chapter 3. A simple model -- Chapter 4. Statement of rigorous results -- Chapter 5. Notation and definitions, resonances -- Chapter 6. Geometric features of the unperturbed problem -- Chapter 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds -- 7.1. Explicit calculations of the perturbed invariant manifold -- Chapter 8. The dynamics in Ã[sub(ε)] -- 8.1. A system of coordinates for Ã[sub(ε)] -- 8.2. Calculation of the reduced Hamiltonian -- 8.3. Isolating the resonances (resonant averaging) -- 8.3.1. The infinitesimal equations for averaging -- 8.3.2. The main averaging result, Theorem 8.9 -- 8.3.3. Proof of Theorem 8.9 -- 8.4. The non-resonant region (KAM theorem) -- 8.4.1. Some results on Diophantine approximation -- 8.4.2. The KAM Theorem for twist maps -- 8.5. Analyzing the resonances -- 8.5.1. Resonances of order 3 and higher -- 8.5.2. Preliminary analysis of resonances of order one or two -- 8.5.3. Primary and secondary tori near the first and second order resonances -- 8.5.4. Proof of Theorem 8.30 and Corollary 8.31 -- 8.5.5. Existence of stable and unstable manifolds of periodic orbits -- Chapter 9. The scattering map -- 9.1. Some generalities about the scattering map -- 9.2. The scattering map in our model: definition and computation -- Chapter 10. Existence of transition chains -- 10.1. Transition chains -- 10.2. The scattering map and the transversality of heteroclinic intersections -- 10.2.1. The non-resonant region and resonances of order 3 and higher -- 10.2.2. Resonances of first order -- 10.2.3. Resonances of order 2; 10.3. Existence of transition chains to objects of different topological types -- Chapter 11. Orbits shadowing the transition chains and proof of theorem 4.1 -- Chapter 12. Conclusions and remarks -- 12.1. The role of secondary tori and the speed of diffusion -- 12.2. Comparison with [DLS00] -- 12.3. Heuristics on the genericity properties of the hypothesis and the phenomena -- 12.4. The hypothesis of polynomial perturbations -- 12.5. Involving other objects -- 12.6. Variational methods -- 12.7. Diffusion times -- Chapter 13. An example -- Acknowledgments -- Bibliography
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ER  -