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 20]. For infinite-dimensional maps and periodic systems, such a proof was completed by Steinlein and Walther in 1990 30], and Henry in 1994 9]. For finite-dimensional autonomous systems, such a proof was accomplished by Palmer in 1996 17]. 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.