2022/10/05 by M. M. Bosschaert, Bosschaert, M. M., Yuri A. Kuznetsov +1 · 1 citation
Computer Science · Mathematics · #34K16 #34K18 #34K19 #37G05 #37G10 #65P30 #Dynamical Systems (math.DS) #FOS: Mathematics #Fractional Differential Equations Solutions #Nonlinear Dynamics and Pattern Formation #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.2210.02560
openalex publication_date 2022/10/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 (DDEs). Using a generalization of the Lindstedt-Poincaré method to approximate the homoclinic solution allows us 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. The effectiveness is demonstrated on various models.