# Hopf-Like Bifurcation in a Wave Equation at a Removable Singularity

## TL;DR

This paper demonstrates a Hopf-like bifurcation in a damped wave equation with a singularity, showing that the bifurcation occurs without the typical small divisor issues by employing a contraction mapping approach.

## Contribution

It introduces a novel method to handle a wave equation bifurcation at a removable singularity without Diophantine conditions or advanced convergence schemes.

## Key findings

- Existence of small amplitude bifurcating periodic solutions
- Removal of small divisor problem at criticality
- Application of contraction mapping principle to wave bifurcation

## Abstract

It is shown that a one-dimensional damped wave equation with an odd time derivative nonlinearity exhibits small amplitude bifurcating time periodic solutions, when the bifurcation parameter is the linear damping coefficient is positive and accumulates to zero. The upshot is that the singularity of the linearized operator at criticality which stems from the well known small divisor problem for the wave operator, is entirely removed without the need to exclude parameters via Diophantine conditions, nor the use of accelerated convergence schemes. Only the contraction mapping principle is used.