Bifurcation analysis of Bogdanov-Takens bifurcations in delay differential equations

TL;DR

This paper analyzes bifurcations in delay differential equations near Bogdanov-Takens points, deriving explicit formulas for bifurcation curves using advanced mathematical techniques and implementing them in DDE-BifTool software.

Contribution

It introduces a novel center manifold reduction and normal form derivation for delay differential equations at Bogdanov-Takens bifurcations, with explicit formulas implemented in software.

Findings

Explicit formulas for bifurcation curves derived

Implementation in DDE-BifTool demonstrated effectiveness

Applicable to various delay differential equation models

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 (DDEs). Using a generalization of the Lindstedt-Poincar\'e 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.