# Nonintegrability of the unfoldings of codimension-two bifurcations

## TL;DR

This paper investigates the nonintegrability of unfoldings of key codimension-two bifurcations in dynamical systems, providing criteria and methods that extend to symmetric systems.

## Contribution

It offers new sufficient conditions for nonintegrability of unfoldings of Fold-Hopf and double-Hopf bifurcations, reducing complex problems to planar polynomial vector fields analysis.

## Key findings

- Derived criteria for nonintegrability of planar polynomial vector fields.
- Reduced bifurcation unfolding problems to planar vector field analysis.
- Applicable to circular symmetric systems and other problems.

## Abstract

Codimension-two bifurcations are fundamental and interesting phenomena in dynamical systems. Fold-Hopf and double-Hopf bifurcations are the most important among them. We study the unfoldings of these two codimension-two bifurcations, and obtain sufficient conditions for their nonintegrability in the meaning of Bogoyavlenskij. We reduce the problems of the unfoldings to those of planar polynomial vector fields and analyze the nonintegrability of the planar vector fields, based on Ayoul and Zung's version of the Morales-Ramis theory. New useful criteria for nonintegrability of planar polynomial vector fields are also given. The approaches used here are applicable to many problems including circular symmetric systems.