The almost-periodic solutions of the weakly coupled pendulum equations

Abstract

Abstract 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, $$\frac{d^{2} x_{n}}{d t^{2}}+\lambda_{n}^{2} \sin x_{n}= \epsilon W_{n}(x_{n-1},x_{n},x_{n-1}),\quad n \in\mathbb {Z}, $$ there are many (positive Lebesgue measure) normally hyperbolic invariant tori which are infinite dimensional in both tangent and normal directions.

Keywords

• Coupled pendulum equations
• Normally hyperbolic invariant tori
• Almost-periodic solutions
• KAM theory