# Boundedness in forced isochronous oscillators

## TL;DR

This paper proves the boundedness of all solutions for a class of forced isochronous oscillators with bounded perturbations and periodic forcing, using advanced twist theorems in both resonant and non-resonant cases.

## Contribution

It extends the application of resonant and non-resonant small twist theorems to establish boundedness in forced isochronous oscillators with general bounded perturbations.

## Key findings

- All solutions are bounded under specified conditions.
- Boundedness holds in both resonant and non-resonant cases.
- The results apply to oscillators with $T$-isochronous potentials.

## Abstract

In this paper we are concerned with the boundedness of all solutions for the forced isochronous oscillator $$x''+V'(x)+g(x)=f(t),$$ where $V$ is a so-called $T$-isochronous potential, the perturbation $g$ is assumed to be bounded, and the $2\pi$-periodic function $f(t)$ is smooth. Using the resonant small twist theorem and averaged small twist theorem established by Ortega, we will prove the boundedness of all solutions for the above forced isochronous oscillator in the resonant and non-resonant cases under some reasonable assumptions, respectively.