Topological methods in the instability problem of Hamiltonian systems
Pages: 295 - 328,
Volume 14, Issue 2,
Full Text (542.9K)
Marian Gidea - Department of Mathematics, Northeastern Illinois University, Chicago, IL 60625, United States (email)
Rafael De La Llave - Department of Mathematics, 1 University Station C1200, University of Texas, Austin, TX 78712, United States (email)
We use topological methods to investigate some recently proposed
mechanisms of instability (Arnol'd diffusion) in Hamiltonian
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
Keywords: Instability, Arnold diffusion, correctly aligned windows.
Mathematics Subject Classification: Primary: 37J40; 37C50; 37C29; 34C37; Secondary:
Received: February 2005;
Published: November 2005.