The large twist theorem and boundedness of solutions for polynomial potentials with $C^1$ time dependent coefficients

In this paper we first prove the so-called large twist theorem, then using it to prove the boundedness of all solutions and the existence of quasi-periodic solutions for Duffing’s equation $$\ddot{x}+x^{2n+1}+{\text{dsum}}_{i=0}^{2n}{p}_i(t)x^i=0,$$ where $p_i(t)\in C^1($\mathbb{S}$) (n+1\leq i\leq 2n)$and$p_i(t)\in C^0($\mathbb{S}$) (0\leq i\leq n)$with$$\mathbb{S}$=$\mathbb{R}$/$\mathbb{Z}$$.