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

## TL;DR

This paper proves a large twist theorem and uses it to establish the boundedness of solutions and existence of quasi-periodic solutions for a class of polynomial Duffing equations with time-dependent coefficients.

## Contribution

It introduces a large twist theorem and applies it to demonstrate boundedness and quasi-periodic solutions for polynomial Duffing equations with $C^1$ and $C^0$ time-dependent coefficients.

## Key findings

- All solutions are bounded under the given conditions.
- Existence of quasi-periodic solutions is established.
- The large twist theorem is proved and applied to nonlinear oscillators.

## Abstract

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 $\mathbb{S}=\mathbb{R}/\mathbb{Z}$.