# Global Hopf bifurcation for differential-algebraic equations with state dependent delay

## TL;DR

This paper develops a global Hopf bifurcation theory for differential-algebraic equations with state-dependent delays, applying it to genetic regulatory models to analyze periodic oscillations.

## Contribution

It introduces a novel global bifurcation framework for differential-algebraic equations with state-dependent delays using $S^1$-equivariant degree, with applications to biological models.

## Key findings

- Established a global Hopf bifurcation theory for these equations.
- Applied the theory to a genetic regulatory model with delay.
- Described the continuation of periodic oscillations in the model.

## Abstract

We develop a global Hopf bifurcation theory for differential equations with a state-dependent delay governed by an algebraic equation, using the $S^1$-equivariant degree. We apply the global Hopf bifurcation theory to a model of genetic regulatory dynamics with threshold type state-dependent delay vanishing at the stationary state, for a description of the global continuation of the periodic oscillations.