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Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions
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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