Topological shadowing methods in arnold diffusion: weak torsion and multiple time scales

Consider a map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carr...

Full description

Saved in:
Bibliographic Details
Published in:Nonlinearity 2023-01, Vol.36 (1), p.426-457
Main Authors: Clarke, Andrew, Fejoz, Jacques, Guàrdia, Marcel
Format: Article
Language:English
Subjects:
37J40
Arnold diffusion
celestial mechanics
Dynamical Systems
Hamiltonian dynamical systems
Mathematics
Citations: Items that this one cites
Items that cite this one
Online Access:Get full text
Tags: Add Tag
No Tags, Be the first to tag this record!
Description
Summary:Consider a map which possesses a normally hyperbolic invariant manifold of any even dimension with transverse homoclinic channels. We develop a topological shadowing argument to prove the existence of Arnold diffusion along the invariant manifold, shadowing some iterations of the inner dynamics carried by the invariant manifold and the outer dynamics induced by the stable and unstable foliations. In doing so, we generalise an idea of Gidea and de la Llave (2006 Discrete Contin. Dyn. Syst. 14 295), based on the method of correctly aligned windows and a so-called transversality-torsion argument. Our proof permits that the dynamics on the invariant manifold satisfy only a non-uniform twist condition, and, most importantly for applications, that the splitting of separatrices be small in certain directions and thus the associated drift in actions very slow; diffusion occurs in the directions of the manifold having non-small splitting. Furthermore we provide estimates for the diffusion time.
ISSN:0951-7715
1361-6544
DOI:10.1088/1361-6544/aca5df