Switching to nonhyperbolic cycles from codimension two bifurcations of equilibria of delay differential equations

Maikel M. Bosschaert, Sebastiaan G. Janssens, Yu A. Kuznetsov

Research output: Contribution to journalArticleAcademicpeer-review

2 Citations (Scopus)

Abstract

In this paper, we perform the parameter-dependent center manifold reduction near the generalized Hopf (Bautin), fold-Hopf, Hopf-Hopf, and transcritical-Hopf bifurcations in delay differential equations (DDEs). This allows us to initialize the continuation of codimension one equilibria and cycle bifurcations emanating from these codimension two bifurcation points. The normal form coefficients are derived in the functional analytic perturbation framework for dual semigroups (sun-star calculus) using a normalization technique based on the Fredholm alternative. The obtained expressions give explicit formulas which have been implemented in the freely available numerical software package DDE-BifTool. While our theoretical results are proven to apply more generally, the software implementation and examples focus on DDEs with finitely many discrete delays. Together with the continuation capabilities of DDE-BifTool, this provides a powerful tool to study the dynamics near equilibria of such DDEs. The effectiveness is demonstrated on various models.

Original languageEnglish
Pages (from-to)252-303
Number of pages52
JournalSIAM journal on applied dynamical systems
Volume19
Issue number1
DOIs
Publication statusPublished - 2020

Keywords

  • Codimension two bifurcations
  • Continuation
  • Delay differential equations
  • Dual perturbation theory
  • Nonhyperbolic cycles
  • Sun-star calculus

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