KAM theorem with large twist and finite smooth large perturbation

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.