KAM theorem for reversible mapping of low smoothness with application

Abstract

Assume the mapping A :is reversible with respect to G : (x, y) → (−x, y), andThen when ε 0 = ε 0 (d) > 0 is small enough and ω is Diophantine, the map A possesses an invariant torus with rotational frequency ω. As an application of the obtained theorem, the Lagrange stability is proved for a class of reversible Duffing equation with finite smooth perturbation.

KAM theorem with large twist and finite smooth large perturbation

“…Then, by (3.10), (3.19)-(3.21), using Rüssmann [27,28] subtle arguments to give optimal estimates of small divisor series (also see Lemma 5.1 in [29]), we get…”