Numerical computation of stability and detection of Hopf bifurcations of steady state solutions of delay differential equations
This paper is concerned with the stability of steady state solutions (fixed points) of systems of time dependent delay differential equations \[ \dot x= f(x(t), x(t-\tau_1),\dots, x(t-\tau_m)), \] with \(x\in\mathbb{R}^n\), and one or several fixed delays \(\tau_i\in \mathbb{R}^+_0\), \(j= 1,\dots, m\). The authors present a robust numerical method to compute the rightmost roots of the nonlinear characteristic equation to the fixed point under consideration. These roots determine the stability of the fixed point, provide inside into the system's behaviour, and can be used for bifurcation detection and indirect calculation of bifurcation points. Examples are presented to demonstrate the usefulness of the method. (One of the delay differential systems is a discrete form of a model equation for the flow of a viscoelastic fluid with fading memory).
scientific article; zbMATH DE number 1309405