# Local bifurcations in differential equations with state-dependent delay

## TL;DR

This paper extends normal form algorithms for delay differential equations to include state-dependent delays, providing a framework for analyzing local bifurcations in more complex dynamical systems.

## Contribution

It develops methods to analyze bifurcations in sd-DDEs based on existing algorithms for constant delays, addressing regularity issues and invariant manifolds.

## Key findings

- Normal form algorithms can be extended to sd-DDEs.
- Invariant manifolds predicted by normal forms persist in full sd-DDEs.
- Partial regularity results support bifurcation analysis in sd-DDEs.

## Abstract

A common task when analysing dynamical systems is the determination of normal forms near local bifurcations of equilibria. As most of these normal forms have been classified and analysed, finding which particular class of normal form one encounters in a numerical bifurcation study guides follow-up computations. This paper builds on normal form algorithms for equilibria of delay differential equations with constant delay that were developed and implemented in DDE-Biftool recently. We show how one can extend these methods to delay-differential equations with state-dependent delay (sd-DDEs). Since higher degrees of regularity of local center manifolds are still open for sd-DDEs, we give an independent (still only partial) argument which phenomena from the truncated normal must persist in the full sd-DDE. In particular, we show that all invariant manifolds with a sufficient degree of normal hyperbolicity predicted by the normal form exist also in the full sd-DDE.