Chaos and shadowing around a homoclinic tube. (English) Zbl 1064.37059

Li, Yanguang

Abstr. Appl. Anal. 2003, No. 16, 923-931 (2003).

Summary: Let \(F\) be a \(C^3\) 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 L. P. Sil’nikov [Sov. Math., Dokl. 9, 624–628 (1968; Zbl 0185.17201)]. Also, the result of Sil’nikov 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 Sil’nikov was done in \(\mathbb{R}^n\) and the current work is done in a Banach space.

MSC:

37L05	General theory of infinite-dimensional dissipative dynamical systems, nonlinear semigroups, evolution equations
37C50	Approximate trajectories (pseudotrajectories, shadowing, etc.) in smooth dynamics
37D45	Strange attractors, chaotic dynamics of systems with hyperbolic behavior
37B10	Symbolic dynamics

Keywords:

diffeomorphism on a Banach space; Bernoulli shift dynamics; shadowing lemma

Citations:

Zbl 0185.17201