# Delay Equations

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#### Critical normal forms for delay differential equations

This is the updated version of: S.G. Janssens, On a Normalization Technique for Codimension Two Bifurcations of Equilibria of Delay Differential Equations, Master Thesis, Utrecht University, 2010. (Latest update: 18 May 2018.)

It was written under supervision of Yu.A. Kuznetsov with Odo Diekmann as co-supervisor. This updated version contains corrections and some new material – see the Summary of updates and corrections. As specified inside the document, it is available under a Creative Commons BY-NC-ND 4.0 International license.

#### The DelayTools package for Maple

The code that was originally part of the thesis text is being rewritten and integrated in the DelayTools package for Maple.

#### Numerical implementation in MATLAB

The newest versions of the software package DDE-BIFTOOL – maintained on SourceForge – is capable of calculating criticial normal form coefficients for local bifurcations, for the case of delay differential equations with finitely many constant delays,
$\dot{x}(t) = f(x(t), x(t – \tau_1),\ldots,x(t – \tau_m),\alpha)$
where $x(t) \in \mathbb{R}^n$, $\alpha \in \mathbb{R}^p$ is a parameter vector and $0 < \tau_1 < \ldots < \tau_m < \infty$. This software implementation is done by Yu.A. Kuznetsov (Utrecht, The Netherlands) and M.M. Bosschaert (Hasselt, Belgium).