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 $$\ddot{x}+x^{2n+1}+\dsum_{i=0}^{2n}p_i(t)x^i=0,$$ 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 $\mathbb{S}=\mathbb{R}/\mathbb{Z}$.