Preservation of Takens-Bogdanov bifurcations for delay differential equations by Euler discretization

TL;DR

This paper investigates how the forward Euler method preserves the Takens-Bogdanov bifurcation structure in delay differential equations, showing that key dynamical features are retained with step size adjustments.

Contribution

It introduces a new technique for calculating normal forms of parameterized maps and demonstrates the preservation of Takens-Bogdanov bifurcations under Euler discretization.

Findings

Takens-Bogdanov point is inherited by Euler method as a 1:1 resonance point.

Hopf and homoclinic branches are shifted by the step size.

Numerical experiments confirm theoretical predictions.

Abstract

A new technique for calculating the normal forms associated with the map restricted to the center manifold of a class of parameterized maps near the fixed point is given first. Then we show the Takens-Bogdanov point of delay differential equations is inherited by the forward Euler method without any shift and turns into a 1:1 resonance point. The normal form near the 1:1 resonance point for the numerical discretization is calculated next by applying the new technique to the map defined by the forward Euler method. The local dynamical behaviors are analyzed in detail through the normal form. It shows the Hopf point branch and the homoclinic branch emanating from the Takens-Bogdanov point are $O(\varepsilon)$ shifted by the forward Euler method, where $\varepsilon$ is step size. At last, a numerical experiment is carried to show the results.