Continuation and bifurcation analysis of delay differential equations 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. par 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. par 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.

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  1. Terrien, Soizic; Krauskopf, Bernd; Broderick, Neil G.R.: Bifurcation analysis of the Yamada model for a pulsing semiconductor laser with saturable absorber and delayed optical feedback (2017)
  2. Gomez, Marcella M.; Sadeghpour, Mehdi; Bennett, Matthew R.; Orosz, Gábor; Murray, Richard M.: Stability of systems with stochastic delays and applications to genetic regulatory networks (2016)
  3. Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire: Investigating irregular behavior in a model for the El Niño southern oscillation with positive and negative delayed feedback (2016)
  4. Maset, Stefano: An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations (2016)
  5. Avedisov, Sergei S.; Orosz, Gábor: Nonlinear network modes in cyclic systems with applications to connected vehicles (2015)
  6. Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire: Delayed feedback versus seasonal forcing: resonance phenomena in an El Niño Southern Oscillation model (2015)
  7. Meijer, Hil G.E.; Coombes, Stephen: Travelling waves in a neural field model with refractoriness (2014)
  8. Orosz, Gábor: Decomposing the dynamics of delayed Hodgkin-Huxley neurons (2014)
  9. Xu, Yingxiang; Mabonzo, Vital D.: Analysis on Takens-Bogdanov points for delay differential equations (2012)
  10. Guo, Songtao; Deng, Shaojiang; Liu, Defang: Hopf and resonant double Hopf bifurcation in congestion control algorithm with heterogeneous delays (2010)
  11. Xu, Jian; Chung, Kwok-Wai: Dynamics for a class of nonlinear systems with time delay (2009)
  12. Roose, Dirk; Szalai, Robert: Continuation and bifurcation analysis of delay differential equations (2007)
  13. Verheyden, Koen; Lust, Kurt: A Newton-Picard collocation method for periodic solutions of delay differential equations (2005)
  14. Balanov, A. G.; Janson, N. B.; McClintock, P. V. E.; Tucker, R. W.; Wang, C. H. T.: Bifurcation analysis of a neutral delay differential equation modelling the torsional motion of a driven drill-string. (2003)
  15. Baker, Christopher T.H.: Retarded differential equations (2000)
  16. Engelborghs, Koen; Luzyanina, Tatyana; Roose, Dirk: Numerical bifurcation analysis of delay differential equations (2000)
  17. Ford, Neville J.; Wulf, Volker: How do numerical methods perform for delay differential equations undergoing a Hopf bifurcation? (2000)
  18. Doedel, Eusebius J.: Nonlinear numerics (1997)
  19. Luzyanina, Tatyana; Engelborghs, Koen; Lust, Kurt; Roose, Dirk: Computation, continuation and bifurcation analysis of periodic solutions of delay differential equations (1997)