# A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model

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*Amadeu Delshams; Rafael de la Llave; Tere M. Seara*

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.

#### Table of Contents

# Table of Contents

## A Geometric Mechanism for Diffusion in Hamiltonian Systems Overcoming the Large Gap Problem: Heuristics and Rigorous Verification on a Model

- Contents v6 free
- Chapter 1. Introduction 110 free
- Chapter 2. Heuristic discussion of the mechanism 716 free
- Chapter 3. A simple model 1524
- Chapter 4. Statement of rigorous results 1928
- Chapter 5. Notation and definitions, resonances 2534
- Chapter 6. Geometric features of the unperturbed problem 2736
- Chapter 7. Persistence of the normally hyperbolic invariant manifold and its stable and unstable manifolds 3140
- Chapter 8. The dynamics in Ã[sub(ε)] 3746
- 8.1. A system of coordinates for Ã[sub(ε)] 3948
- 8.2. Calculation of the reduced Hamiltonian 4150
- 8.3. Isolating the resonances (resonant averaging) 4352
- 8.4. The non-resonant region (KAM theorem) 5059
- 8.5. Analyzing the resonances 5867
- 8.5.1. Resonances of order 3 and higher 5867
- 8.5.2. Preliminary analysis of resonances of order one or two 5968
- 8.5.3. Primary and secondary tori near the first and second order resonances 6271
- 8.5.4. Proof of Theorem 8.30 and Corollary 8.31 6877
- 8.5.5. Existence of stable and unstable manifolds of periodic orbits 8291

- Chapter 9. The scattering map 8796
- Chapter 10. Existence of transition chains 97106
- Chapter 11. Orbits shadowing the transition chains and proof of theorem 4.1 121130
- Chapter 12. Conclusions and remarks 123132
- 12.1. The role of secondary tori and the speed of diffusion 123132
- 12.2. Comparison with [DLS00] 123132
- 12.3. Heuristics on the genericity properties of the hypothesis and the phenomena 124133
- 12.4. The hypothesis of polynomial perturbations 125134
- 12.5. Involving other objects 126135
- 12.6. Variational methods 127136
- 12.7. Diffusion times 127136

- Chapter 13. An example 129138
- Acknowledgments 135144
- Bibliography 137146