KAM theorem for reversible mapping of low smoothness with application

Abstract

Assume the mapping $$A:\{\begin{array}{l}{x}_1=x+\omega +y+f(x,y),y_1=y+g(x,y),\end{array}(x,y)\in {\mathbb{T}}^d\times B(r_0)$$ is reversible with respect to $G:(x,y)\mapsto (-x,y),$ and $|f{|}_{C^{\ell }({\mathbb{T}}^d\times B(r_0))}\le {\epsilon }_0,|g{|}_{C^{\ell +d}({\mathbb{T}}^d\times B(r_0))}\le {\epsilon }_0,$ where $B(r_0):=\{|y|\le r_0:y\in {\mathbb{R}}^d\},$ $\ell =2d+1+\mu$ with $0<\mu \ll 1.$ Then when ${\epsilon }_0={\epsilon }_0(d)>0$ 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.