Abstract. 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. 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. Contents
|
105
|
Les m'ethodes nouvelles de la m'ecanique c'eleste
– Poincar'e
- 1892
|
|
79
|
Instability of dynamical systems with several degrees of freedom
– Arnold
- 1964
|
|
75
|
Ergodic Problems of Classical Mechanics
– Arnold, Avez
- 1968
|
|
67
|
Stable and random motions in dynamical systems
– Moser
- 1973
|
|
62
|
Persistence and Smoothness of Invariant Manifolds for Flows
– Fenichel
- 1971
|
|
55
|
Invariant manifolds
– Hirsch, Pugh, et al.
- 1977
|
|
54
|
The separation of motions in systems with rapidly rotating phase
– Neishtadt
- 1984
|
|
53
|
A universal instability of many-dimensional oscillator systems
– Chirikov
- 1979
|
|
42
|
Twistless KAM tori, quasi flat homoclinic intersections, and other cancellations in the perturbation series of certain completely integrable Hamiltonian systems. A review
– Gallavotti
- 1994
|
|
40
|
Integrability of Hamiltonian systems on Cantor sets
– Poschel
- 1982
|
|
38
|
Generalized implicit function theorems with applications to some small divisor problems
– Zehnder
- 1976
|
|
37
|
Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the hamiltonian
– Arnold
- 1963
|
|
33
|
Symplectic maps, variational principles, and transport
– Meiss
- 1992
|
|
30
|
Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations
– Herman
- 1979
|
|
28
|
Asymptotic stability with rate conditions
– Fenichel
- 1977
|
|
28
|
On the conservation of hyperbolic invariant tori for Hamiltonian systems
– Graff
- 1974
|
|
22
|
Sur les courbes invariantes par les difféomorphismes de l’anneau
– Herman
- 1983
|
|
20
|
Variational construction of connecting orbits
– Mather
- 1993
|
|
16
|
Effective stability and
– Delshams, Gutiérrez
- 1996
|
|
16
|
The Splitting of Separatrices for Analytic Diffeomorphisms
– Fontich, Sim'o
- 1990
|
|
16
|
Multiphase Averaging for Classical Systems
– Lochak, Meunier
- 1988
|
|
16
|
diffusion: a variational construction
– Arnold
- 1998
|
|
15
|
Drift and diffusion in phase space
– Chierchia, Gallavotti
- 1994
|
|
14
|
Drift and diffusion in phase
– Chierchia, Gallavotti
- 1994
|
|
14
|
la Llave and
– de
- 1989
|
|
13
|
Melnikov's method and Arnold diffusion for perturbations of integrable Hamiltonian systems
– Holmes, Marsden
- 1982
|
|
13
|
Frequency analysis for multi-dimensional systems. Global dynamics and diffusion
– Laskar
- 1993
|
|
12
|
Stabilité ou instabilité des points fixes elliptiques
– Douady
- 1988
|
|
11
|
A geometric approach to the existence of orbits with unbounded energy in generic periodic perturbations by a potential of generic geodesic flows of T 2
– Delshams, Llave, et al.
|
|
11
|
diffusion in perturbations of analytic integrable Hamiltonian systems
– Arnold
- 1998
|
|
11
|
Chaos Near Resonance
– Haller
- 1999
|
|
10
|
Unbounded growth of energy in nonautonomous Hamiltonian systems
– Bolotin, Treschev
- 1999
|
|
10
|
Lie Transform Perturbation Theory for Hamiltonian Systems
– Cary
- 1981
|
|
10
|
Splitting potential and Poincare-Melnikov method for whiskered tori in Hamiltonian systems
– Delshams, Gutiérrez
- 2000
|
|
9
|
Arnold’s diffusion in isochronous systems
– Gallavotti
- 1998
|
|
9
|
Estimates in the Kolmogorov theorem on conservation of conditionally periodic motions
– Neishtadt
- 1982
|
|
8
|
Upper bounds on Arnold diffusion times via Mather theory
– Bessi, Chierchia, et al.
- 2001
|
|
8
|
de la Llave: A rigorous partial justification of Greene's criterion
– Falcolini, R
- 1992
|
|
7
|
A functional analysis approach to Arnold diffusion
– Berti, P
- 1997
|
|
7
|
Graduate course at Princeton, 95–96, and Lectures at Penn State, Spring 96
– Mather
- 1995
|
|
7
|
Multidimensional symplectic separatrix maps
– Treschev
- 2000
|
|
6
|
Invariant manifolds for near identity differentiable maps and splitting of separatrices. Ergodic Theory Dynam
– Fontich, Simó
- 1990
|
|
6
|
Universal homoclinic bifurcations and chaos near double resonances
– Haller
- 1997
|
|
6
|
Transition tori in the five-body problem
– Moeckel
- 1996
|
|
5
|
Arnol ′ d. Small denominators and problems of stability of motion in classical and celestial mechanics
– I
- 1963
|
|
5
|
Rafael de la Llave. The parameterization method for invariant manifolds III: overview and applications
– Cabré, Fontich
- 2003
|
|
5
|
A theory of modulational diffusion
– Chirikov, Lieberman, et al.
- 1985
|
|
5
|
Homoclinic orbits to invariant tori in Hamiltonian systems
– Delshams, Guti'errez
- 1997
|
|
5
|
Hamiltonian systems with orbits covering densely submanifolds of small codimension
– Fontich, Martín
|
|
5
|
la Llave. A tutorial on KAM theory
– de
- 1999
|
|
