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. Khristichenko, M. Yu.; Nechepurenko, Yu. M.: Computation of periodic solutions to models of infectious disease dynamics and immune response (2021)
  2. Andò, Alessia; Breda, Dimitri: Convergence analysis of collocation methods for computing periodic solutions of retarded functional differential equations (2020)
  3. Gimeno, Joan; Jorba, Àngel: Using automatic differentiation to compute periodic orbits of delay differential equations (2020)
  4. Nechepurenko, Yuri; Khristichenko, Michael; Grebennikov, Dmitry; Bocharov, Gennady: Bistability analysis of virus infection models with time delays (2020)
  5. Randall, E. Benjamin; Randolph, Nicholas Z.; Olufsen, Mette S.: Persistent instability in a nonhomogeneous delay differential equation system of the Valsalva maneuver (2020)
  6. Scholl, T. H.; Gröll, L.; Hagenmeyer, V.: Time delay in the swing equation: a variety of bifurcations (2019)
  7. Keane, A.; Krauskopf, B.: Chenciner bubbles and torus break-up in a periodically forced delay differential equation (2018)
  8. Avedisov, Sergei S.; Orosz, Gábor: Analysis of connected vehicle networks using network-based perturbation techniques (2017)
  9. Calleja, R. C.; Humphries, A. R.; Krauskopf, B.: Resonance phenomena in a scalar delay differential equation with two state-dependent delays (2017)
  10. Molnar, T. G.; Dombovari, Z.; Insperger, T.; Stepan, G.: On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction (2017)
  11. 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)
  12. 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)
  13. 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)
  14. Maset, Stefano: An abstract framework in the numerical solution of boundary value problems for neutral functional differential equations (2016)
  15. Avedisov, Sergei S.; Orosz, Gábor: Nonlinear network modes in cyclic systems with applications to connected vehicles (2015)
  16. Keane, Andrew; Krauskopf, Bernd; Postlethwaite, Claire: Delayed feedback versus seasonal forcing: resonance phenomena in an El Niño Southern Oscillation model (2015)
  17. Słowiński, Piotr; Krauskopf, Bernd; Wieczorek, Sebastian: Mode structure of a semiconductor laser with feedback from two external filters (2015)
  18. Meijer, Hil G. E.; Coombes, Stephen: Travelling waves in a neural field model with refractoriness (2014)
  19. Orosz, Gábor: Decomposing the dynamics of delayed Hodgkin-Huxley neurons (2014)
  20. Xu, Yingxiang; Mabonzo, Vital D.: Analysis on Takens-Bogdanov points for delay differential equations (2012)

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