April  2006, 14(2): 295-328. doi: 10.3934/dcds.2006.14.295

Topological methods in the instability problem of Hamiltonian systems

1. 

Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, United States

2. 

Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States

Received  February 2005 Revised  June 2005 Published  November 2005

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 different topology), (b) the manifolds $W^\s_\Lambda$, $W^\u_\Lambda$ intersect transversally, (c) the systems satisfies 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 differentiability.
Also, the method presented here allows us to produce concrete estimates on the time to move, which were not considered in the previous papers.
Citation: Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete & Continuous Dynamical Systems - A, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295
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