Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

Details Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions

2002-03-13

arxiv.org/abs/nlin/0203024v1

Nonlinear Sciences

Chaotic Dynamics

Scientific paper

10.1023/B:JODY.0000010062.09599.

For finite-dimensional maps and periodic systems, Palmer rigorously proved Smale horseshoe theorem using shadowing lemma in 1988. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990, and Henry in 1994. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996. For infinite-dimensional autonomous systems, the current article offers such a proof. First we prove an Inclination Lemma to set up a coordinate system around a pseudo-orbit. Then we utilize graph transform and the concept of persistence of invariant manifold, to prove the existence of a shadowing orbit.

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