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.