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. It is maintained on GitHub under a BSD 3-clause license.

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).