KAM theorem for reversible mapping of low smoothness with application

TL;DR

This paper proves a KAM theorem for reversible mappings with low smoothness, establishing the existence of invariant tori under small perturbations, and applies it to demonstrate Lagrange stability in certain reversible Duffing equations.

Contribution

It extends KAM theory to low-smoothness reversible mappings and provides an application to reversible Duffing equations, which was previously unexplored.

Findings

Existence of invariant tori under small perturbations for low-smoothness reversible maps.

Extension of KAM theorem to mappings with finite smoothness.

Lagrange stability established for a class of reversible Duffing equations.

Abstract

Assume the mapping $$A:\left\{ \begin{array}{ll} x_{1}=x+\omega+y+f(x,y), y_{1}=y+g(x,y), \end{array} \right. (x, y)\in \mathbb{T}^{d}\times B(r_{0}) $$ is reversible with respect to $G: (x, y)\mapsto (-x, y),$ and $| f | _{C^{\ell}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0}, | g |_{C^{\ell+d}(\mathbb{T}^{d}\times B(r_{0}))}\leq \varepsilon_{0},$ where $B(r_{0}):=\{|y|\le r_0:\; y\in\mathbb R^d\},$ $\ell=2d+1+\mu$ with $0<\mu\ll 1.$ Then when $\varepsilon_{0}=\varepsilon_{0}(d)>0$ is small enough and $\omega$ is Diophantine, the map $A$ possesses an invariantS torus with rotational frequency $\omega.$ As an application of the obtained theorem, the Lagrange stability is proved for a class of reversible Duffing equation with finite smooth perturbation.