Chaos and Shadowing Around a Homoclinic Tube

Details Chaos and Shadowing Around a Homoclinic Tube Chaos and Shadowing Around a Homoclinic Tube

2003-04-25

arxiv.org/abs/nlin/0304050v1

Nonlinear Sciences

Chaotic Dynamics

accepted, Abstract and Applied Analysis

Scientific paper

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 shadowing lemma. This work removes an uncheckable condition of Silnikov [Equation (11), page 625 of L. P. Silnikov, Soviet Math. Dokl., vol.9, no.3, (1968), 624-628]. Also, the result of Silnikov does not imply Bernoulli shift dynamics of a single map, rather only provides a labeling of all invariant tubes around the homoclinic tube. The work of Silnikov was done in ${\mathbb R}^n$, and the current work is done in a Banach space.

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