KAM theorem with large twist and finite smooth large perturbation

TL;DR

This paper proves the persistence of invariant tori in a non-degenerate Hamiltonian system with large twist and finite smooth large perturbation, extending KAM theory to more complex, highly perturbed systems.

Contribution

It extends KAM theorem results to Hamiltonian systems with large twist and finite smooth large perturbations, including applications to Duffing oscillator networks.

Findings

Existence of invariant tori with Diophantine frequencies for small perturbations.

Application to Lagrangian stability in networks of Duffing oscillators.

Extension of KAM theory to systems with large twist and finite smooth perturbations.

Abstract

In the present paper, we will discuss the following non-degenerate Hamiltonian system \begin{equation*} H(\theta,t,I)=\frac{H_0(I)}{\varepsilon^{a}}+\frac{P(\theta,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 $a>b$, $H_0: [1,2]^d\rightarrow \mathbf R$ is real analytic and $P: \mathbf T^{d+1}\times [1,2]^d\rightarrow \mathbf R$ is $C^{\ell}$ with $\ell=\frac{2(d+1)(5a-b+2ad)}{a-b}+\mu$, $0<\mu\ll1$. We prove that if $\varepsilon$ 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.