# Boundedness of solutions for Duffing equation with low regularity in time

## TL;DR

This paper proves that solutions to a Duffing equation with low regularity time-dependent coefficients are bounded, extending understanding of solution behavior under less smooth conditions.

## Contribution

It establishes boundedness of solutions for a Duffing equation with coefficients of low regularity in time, a novel result in the study of nonlinear oscillators.

## Key findings

- All solutions are bounded under specified regularity conditions.
- Boundedness holds even with coefficients of limited smoothness.
- The result broadens the class of Duffing equations with predictable solution behavior.

## Abstract

It is shown that all solutions are bounded for Duffing equation $\ddot{x}+ x^{2n+1}+\sum_{j=0}^{2n}P_{j}(t)x^{j}=0,$ provided that for each $n+1\le j\le 2n$, $P_j(t)\in C^{\gamma}(\mathbb T)$ with $\gamma>1-\frac1n$ and for each $0\le j\le n$, $P_j\in L(\mathbb T^1)$.