# Non-Existence of Periodic Orbits for Forced-Damped Potential Systems in Bounded Domains

## TL;DR

This paper establishes conditions under which forced-damped potential systems in bounded domains cannot have periodic solutions, using Lr-estimates and gradient bounds, with practical examples illustrating the results.

## Contribution

It provides a novel non-existence theorem for periodic solutions in bounded domains for forced-damped systems based on gradient bounds of the potential.

## Key findings

- Non-existence of periodic solutions under certain gradient bounds
- Lr-estimates for periodic solutions of damped systems
- Illustrative examples demonstrating the theoretical results

## Abstract

We prove Lr-estimates on periodic solutions of periodically-forced, linearly-damped mechanical systems with polynomially-bounded potentials. The estimates are applied to obtain a non-existence result of periodic solutions in bounded domains, depending on an upper bound on the gradient of the potential. The results are illustrated on examples.