Topological methods in the instability problem of Hamiltonian systems
type: Item
instance of: scholarly article
title: Topological methods in the instability problem of Hamiltonian systems
zbMATH Open document ID: 1098.37056
author: Marian Gidea
author: Rafael de la Llave
published in: Discrete and Continuous Dynamical Systems
publication date: 2006-07-06T00:00:00Z
review text: We use topological methods to investigate some recently proposed mechanisms of instability (Arnol'd diffusion) in Hamiltonian systems. In these mechanisms, chains of heteroclinic connections between whiskered tori are constructed, based on the existence of a normally hyperbolic manifold \(\Lambda\), so that: (a) the manifold \(\Lambda\) is covered rather densely by transitive tori (possibly of diferent topology), (b) the manifolds \(W_\Lambda^s, W_\Lambda^u\) intersect transversally, (c) the systems satisfy some explicit non-degeneracy assumptions, which hold generically. In this paper, we use the method of correctly aligned windows to show that, under the assumptions (a), (b), (c), there are orbits that move a significant amount. As a matter of fact, the method presented here does not require that the tori are exactly invariant, only that they are approximately invariant. Hence, compared with the previous papers, we do not need to use KAM theory. This lowers the assumptions on diferentiability. Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 37C50
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 34C37
Mathematics Subject Classification ID: 70H08
zbMATH DE Number: 5039054
zbMATH Keywords: instability
zbMATH Keywords: Arnol'd diffusion
zbMATH Keywords: correctly aligned windows
MaRDI profile type: Publication
full work available at URL: dcds.2006.14.295
OpenAlex ID: W2150429134
DOI: 10.3934/DCDS.2006.14.295
description: scientific article