Abstract and Applied Analysis
Volume 2003 (2003), Issue 16, Pages 923-931
doi:10.1155/S1085337503304038
Chaos and shadowing around a homoclinic tube
Department of Mathematics, University of Missouri, Columbia 65211, MO, USA
Received 7 February 2003
Copyright © 2003 Yanguang (Charles) Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
Let F be a C3 diffeomorphism on a Banach space B. F has a homoclinic tube asymptotic to an invariant manifold. Around the
homoclinic tube, Bernoulli shift dynamics of submanifolds is
established through a shadowing lemma. This work removes an
uncheckable condition of Silnikov (1968). Also, the result of Silnikov does not imply
Bernoulli shift dynamics of a single map, but rather only provides a
labeling of all invariant tubes around the homoclinic tube. The
work of Silnikov was done in ℝn and the current work is done in a Banach space.