Remarks on the preservation and breaking of translational symmetry for a class of ODEs

TL;DR

This paper investigates conditions under which translational symmetry is preserved or broken in certain periodic second-order ODEs, extending previous theorems by employing novel group actions to analyze symmetry properties.

Contribution

It introduces new symmetry preservation and breaking theorems for periodic ODEs using unconventional group actions, generalizing prior results by Willem and Costa-Fang.

Findings

Established conditions for symmetry preservation in periodic ODEs.

Proved symmetry breaking scenarios under specific nonlinearities.

Extended existing theorems with broader group action frameworks.

Abstract

In this paper, we provide both a preservation and breaking of symmetry theorem for $2\pi$-periodic problems of the form \begin{align*} \begin{cases} -u''(t) + g(u(t)) = f(t)\cr u(0) - u(2\pi) = u'(0) - u'(2\pi) = 0 \end{cases} \end{align*} where $g: \mathbb{R} \to \mathbb{R}$ is a given $C^1$ function and $f: [0,2\pi] \to \mathbb{R}$ is continuous. We provide a preservation of symmetry result that is analogous to one given by Willem (Willem, 1989) and a generalization of the theorem given by Costa-Fang (Costa and Fang, 2019). Both of these theorems use group actions that are not normally considered in the literature.