Numerical Computation of Takens-Bogdanov Points for Delay Differential Equations

TL;DR

This paper introduces a numerical method for directly computing Takens-Bogdanov points in delay differential equations with two parameters, simplifying the process through algebraic reduction and Newton iteration.

Contribution

It develops a finite-dimensional algebraic approach to accurately compute Takens-Bogdanov points in delay differential systems.

Findings

The method effectively reduces the defining system to a finite algebraic form.

Takens-Bogdanov points can be approximated using standard Newton iteration.

The approach simplifies the computation of bifurcation points in delay systems.

Abstract

The paper presents a numerical technique for computing directly the Takens-Bogdanov points in the nonlinear system of differential equations with one constant delay and two parameters. By representing the delay differential equations as abstract ordinary differential equations in their phase spaces, the quadratic Takens-Bogdanov point is defined and a defining system for it is produced. Based on the descriptions for the eigenspace associated with the double zero eigenvalue, we reduce the defining system to a finite dimensional algebraic equation. The quadratic Takens-Bogdanov point, together with the corresponding values of parameters, is proved to be the regular solution of the reduced defining system and then can be approximated by the standard Newton iteration directly.