A KAM Theorem with Large Twist and Finite Smooth Large Perturbation

Abstract

We study non-degenerate Hamiltonian systems of the form

$$\begin{aligned} H(\theta ,t,I)=\frac{H_0(I)}{\varepsilon ^{a}}+\frac{P(\theta ,t,I)}{\varepsilon ^{b}}, \end{aligned}$$

where $(\theta ,t,I)\in \mathbf {{T}}^{d+1}\times [1,2]^d$ ($\mathbf {{T}}:=\mathbf {{R}}/{2\pi {\textbf{Z}}}$), a, b are given positive constants with $a>b$, $H_0:[1,2]^d\rightarrow \textbf{R}$ is real analytic and $P:{\textbf{T}}^{d+1}\times [1,2]^d\rightarrow {\textbf{R}}$ is $C^{\ell }$ with $\ell =\frac{2(d+1)(5a-b+2ad)}{a-b}+\mu$, $0<\mu \ll 1$. We prove that, for the above Hamiltonian system, if $\varepsilon$ is sufficiently small, there is an invariant torus with given Diophantine frequency vector which obeys conditions (1.7) and (1.8). As for application, a finite network of Duffing oscillators with periodic external forces possesses Lagrange stability for almost all initial data.