On the computation of the term $w_{21}z^2\bar{z}$ of the series defining the center manifold for a scalar delay differential equation

TL;DR

This paper addresses the calculation of specific third-order terms in the center manifold series for scalar delay differential equations, resolving issues of non-uniqueness in the algebraic system involved.

Contribution

It identifies the dependency problem in the algebraic system for computing the term $w_{21}$ and provides a method to obtain a unique formula for $w_{21}(0)$.

Findings

The algebraic system for $w_{21}$ is dependent, leading to infinitely many solutions.

A method to overcome the non-uniqueness and compute a specific value of $w_{21}(0)$.

Clarification of the structure of the equations involved in center manifold computations.

Abstract

In computing the third order terms of the series of powers of the center manifold at an equilibrium point of a scalar delay differential equation, with a single constant delay $r>0,$ some problems occur at the term $w_{21}z^2\bar{z}.$ More precisely, in order to determine the values at 0, respectively $-r$ of the function $w_{21}(\,.\,),$ an algebraic system of equations must be solved. We show that the two equations are dependent, hence the system has an infinity of solutions. Then we show how we can overcome this lack of uniqueness and provide a formula for $w_{21}(0).$