Invariant curves of low smooth quasi-periodic reversible mappings

TL;DR

This paper develops a new method to find invariant curves in low smoothness quasi-periodic reversible mappings by linking them to reversible differential equations and establishing a KAM theorem for such systems.

Contribution

It introduces a novel approach using reversible differential equations and a KAM theorem to obtain invariant curves, overcoming limitations of previous approximation methods.

Findings

Established invariant curves for low smoothness reversible mappings.

Developed variants of invariant curve theorems for quasi-periodic reversible mappings.

Extended the applicability of KAM theory to less smooth reversible systems.

Abstract

In this paper, we obtain the invariant curves of quasi-periodic reversible mappings with finite smoothness. Since the reversible property is difficult to maintain in the process of approximating smooth functions by analytical ones, R\"{u}ssmann's method in \cite{HR} is invalid. Inspired by the recent work of Li, Qi and Yuan in \cite{LJ}, we turn to regard the reversible mapping as the Poincar\'{e} map of a reversible differential equation. By constructing a KAM theorem for a reversible differential equation which is quasi-periodic in time, we obtain the invariant curves of the reversible mapping. Beyond that, we establish some variants of invariant curve theorems for quasi-periodic reversible mappings.