Bifurcation Analysis of Bogdanov-Takens Bifurcations in Delay Differential Equations

Maikel Bosschaert, Yu.A. Kuznetsov

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

In this paper, we will perform the parameter-dependent center manifold reduction near the generic and transcritical codimension two Bogdanov-Takens bifurcation in classical delay differential equations. Using an approximation to the homoclinic solutions derived with a generalized Lindstedt-Poincar\'e method, we develop a method to initialize the continuation of the homoclinic bifurcation curves emanating from these points. The normal form transformation is 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 bifurcation software package DDE-BifTool.

Original languageEnglish
Pages (from-to)553-591
Number of pages39
JournalSIAM Journal on Applied Dynamical Systems
Volume23
Issue number1
DOIs
Publication statusPublished - 30 Jan 2024

Keywords

  • Bogdanov-Takens bifurcation
  • Center manifold theorem
  • DDE-BifTool
  • Delay differential equations
  • Strongly continuous semigroups
  • Sun-star calculus
  • homoclinic solutions
  • delay differential equations
  • generic Bogdanov-Takens bifurcation
  • transcritical Bogdanov-Takens bifurcation
  • strongly continuous semigroups
  • center manifold theorem
  • sun-star calculus

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