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For finite-dimensional maps and periodic systems, Palmer rigorously provedSmale horseshoe theorem using shadowing lemma in 1988. For infinite-dimensionalmaps and periodic systems, such a proof was completed by Steinlein and Waltherin 1990, and Henry in 1994. For finite-dimensional autonomous systems, such aproof was accomplished by Palmer in 1996. For infinite-dimensional autonomoussystems, the current article offers such a proof. First we prove an InclinationLemma to set up a coordinate system around a pseudo-orbit. Then we utilizegraph transform and the concept of persistence of invariant manifold, to provethe existence of a shadowing orbit

Chaos and Shadowing Lemma for Autonomous Systems of Infinite Dimensions