Boundedness of solutions in impulsive Duffing equations with polynomial potentials and $C^{1}$ time dependent coefficients

Abstract

In this paper, we are concerned with the impulsive Duffing equation $$ x''+x^{2n+1}+\sum_{i=0}^{2n}x^{i}p_{i}(t)=0,\ t\neq t_{j}, $$ with impulsive effects $x(t_{j}+)=x(t_{j}-),\ x'(t_{j}+)=-x'(t_{j}-),\ j=\pm1,\pm2,\cdots$, where the time dependent coefficients $p_i(t)\in C^1(\mathbb{S}^1)\ (n+1\leq i\leq 2n)$ and $p_i(t)\in C^0(\mathbb{S}^1)\ (0\leq i\leq n)$ with $\mathbb{S}^1=\mathbb{R}/\mathbb{Z}$. If impulsive times are 1-periodic and $t_{2}-t_{1}\neq\frac{1}{2}$ for $0< t_{1}<t_{2}<1$, basing on a so-called large twist theorem recently established by X. Li, B. Liu and Y. Sun in \cite{XLi}, we find large invariant curves diffeomorphism to circles surrounding the origin and going to infinity, which confines the solutions in its interior and therefore leads to the boundedness of these solutions. Meanwhile, it turns out that the solutions starting at $t=0$ on the invariant curves are quasiperiodic.