Continuation and bifurcation analysis of delay differential equations. (English) Zbl 1132.34001

Krauskopf, Bernd (ed.) et al., Numerical continuation methods for dynamical systems. Path following and boundary value problems. Dordrecht: Springer (ISBN 978-1-4020-6355-8/hbk). Understanding Complex Systems, 359-399 (2007).
This is a review article on numerical continuation and bifurcation analysis methods for delay differential equations (DDEs). The article is not meant as an introduction into the theory of DDEs or numerical methods for dynamical systems but it is ideal as an entry point for further reference.
Both authors are directly linked to the two currently freely available numerical packages for this purpose, PDDE-CONT and DDE-BIFTOOL. One author, R. Szalai, is the author and maintainer of PDDE-CONT, the other author, D. Roose, is the initiator (together with K. Engelborghs and T. Luzyanina) of DDE-BIFTOOL. The review discusses the numerical approaches implemented in PDDE-CONT and DDE-BIFTOOL for computation and stability analysis of equilibria and periodic orbits, and for continuing codimension-one bifurcations of equilibria and periodic orbits.
In its second part the review covers more specialist topics such as the computation and continuation of connecting orbits or quasi-periodic tori, and the treatment of neutral equations or state-dependent delays. The authors also discuss briefly, and refer to, related or alternative approaches by others (specifically, work by Breda and Barton).
The paper is rounded off by several complex examples: a semiconductor laser with feedback (an example which has a continuous symmetry), a high-dimensional traffic model, a (regularized) nonsmooth model for chattering, and the classical Mackey-Glass equation.
For the entire collection see [Zbl 1117.65005].


34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations
34K18 Bifurcation theory of functional-differential equations
34K28 Numerical approximation of solutions of functional-differential equations (MSC2010)