KAM theorem with large twist and finite smooth large perturbation

Abstract

In the present paper, we will discuss the following non-degenerate Hamiltonian system \begin{equation*} H(θ,t,I)=\frac{H_0(I)}{\varepsilon^{a}}+\frac{P(θ,t,I)}{\varepsilon^{b}}, \end{equation*} where $(\theta ,t,I)\in {\mathbf{T}}^{d+1}\times [1,2{]}^d$ ($\mathbf{T}:=\mathbf{R}/2\pi \mathbf{Z}$), $a,b$ are given positive constants with $a>b$, $H_0:[1,2{]}^d\to \mathbf{R}$ is real analytic and $P:{\mathbf{T}}^{d+1}\times [1,2{]}^d\to \mathbf{R}$ is $C^{\ell }$ with $\ell =\frac{2(d+1)(5a-b+2ad)}{a-b}+\mu$, $0<\mu \ll 1$. We prove that if $\epsilon$ is sufficiently small, there is an invariant torus with given Diophantine frequency vector for the above Hamiltonian system. As for application, we prove that a finite network of Duffing oscillators with periodic exterior forces possesses Lagrangian stability for almost all initial data.

References in corpus (3)

• A lecture on the classical KAM theorem
• Resonance Clustering in Globally Coupled Electrochemical Oscillators with External Forcing
• KAM theorem with large perturbation and application to network of Duffing oscillators