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Abstract and Applied Analysis
Volume 2003 (2003), Issue 16, Pages 923-931

Chaos and shadowing around a homoclinic tube

Department of Mathematics, University of Missouri, Columbia 65211, MO, USA

Received 7 February 2003

Copyright © 2003 Hindawi Publishing Corporation. 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.


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 n and the current work is done in a Banach space.