A geometric mechanism for diffusion in Hamiltonian systems overcoming the large gap problem: heuristics and rigorous verification on a model

Abstract

Chapter 1. Introduction 1 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 inΛ ε 37 8.1. A system of coordinates forΛ ε 39 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 62 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 v vi CONTENTS Chapter 10. Existence of transition chains 97 10.1. Transition chains 99 10.2. The scattering map and the transversality of heteroclinic intersections 99 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

Geography of resonances and Arnold diffusion ina prioriunstable Hamiltonian systems

“…In this paper we perform an accurate process of truncation of the Fourier series of the perturbation and we present a deeper study of the geography of resonances. Using this, we are able to extend and simplify some of the results in [DLS06a] and apply them to an a priori unstable Hamiltonian system with a generic perturbation.…”

“…The goal of this paper is to present a generalization of the geometric mechanism for global instability (popularly known as Arnold diffusion) in a priori unstable Hamiltonian systems introduced in [DLS06a]. That paper developed an argument to prove the existence of global instability in a-priori unstable nearly integrable Hamiltonian systems (the unperturbed Hamiltonian presents hyperbolicity, so that it can not be expressed globally in action-angle variables) and applied it to a model which presented the so called large gap problem.…”

“…Of particular interest for the present paper are [DLS00, DLS06a,DLS06b]. The strategy in the mentioned papers is based on the incorporation of new invariant objects, created by the resonances, like secondary KAM tori and the stable and unstable manifolds of lower dimensional tori in the transition chain, together with the primary KAM tori.…”

Arnold Diffusion in A Priori Chaotic Symplectic Maps

“…A system of this type is called a priori unstable. The drift of orbits along the cylinder has been actively studied in the last decade [3][4][5][7][8][9]12,[18][19][20]22,27,28,[31][32][33][34]70,87,88], including the problem of genericity of this phenomenon and instability times. It should be noted that the Arnold diffusion can be much faster in this case.…”

“…The process can be described using the notion of a scattering map introduced by Delshams et al [32]. Earlier Moeckel [75] suggested that Arnold diffusion can be modelled by random application of two area-preserving maps on a cylinder (this approach was recently continued in [17,[46][47][48][49]53,66]).…”