Boundedness of solutions for the reversible system with low regularity in time

Abstract

In the present paper, it is proved that all solutions are bounded for the reversible system \ddot{x}+\sum_{i=0}^{l}b_{i}(t)x^{2i+1}\dot{x}+x^{2n+1}+\sum_{i=0}^{n-1}a_{i}(t)x^{2i+1}=0, 0\leq l\leq [\frac{n}{2}]-1,t\in\mathbb{T}^{1}=\mathbb{R}/\mathbb{Z}, where a_{i}(t)\in C^{1}(\mathbb{T}^{1})\;([\frac{n-1}{2}]+1\leq i\leq n-1), a_{j}(t)\in L^{1}(\mathbb{T}^{1})\;(0\leq j\leq [\frac{n-1}{2}]) and b_{k}(t)\in C^{1}(\mathbb{T}^{1})\;(0\leq k\leq l).