The almost-periodic solutions of the weakly coupled pendulum equations

In this paper, it is proved that, for the networks of weakly coupled pendulum equations d2xndt2+λn2sinxn=ϵWn(xn−1,xn,xn−1),n∈Z,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d^{2} x_{n}}{d t^{2}}+\lambda_{n}^{2} \sin x_{n}= \epsilon W_{n}(x_{n-1},x_{n},x_{n-1}),\quad n \in\mathbb {Z}, $$\end{document} there are many (positive Lebesgue measure) normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions.