Fast generation of stability charts for time-delay systems using continuation of characteristic roots

TL;DR

This paper introduces a novel continuation method for efficiently generating stability charts of delay differential equations by tracking characteristic roots as functions of parameters, significantly reducing computational effort.

Contribution

The paper presents a new continuation of characteristic roots (CCR) method that simplifies stability analysis of DDEs by solving ODEs instead of large eigenvalue problems, enhancing speed and accuracy.

Findings

CCR method is up to 10 times faster than Galerkin approximation.

The method produces highly accurate stability charts.

It reduces computational complexity in stability analysis of DDEs.

Abstract

Many dynamic processes involve time delays, thus their dynamics are governed by delay differential equations (DDEs). Studying the stability of dynamic systems is critical, but analyzing the stability of time-delay systems is challenging because DDEs are infinite-dimensional. We propose a new approach to quickly generate stability charts for DDEs using continuation of characteristic roots (CCR). In our CCR method, the roots of the characteristic equation of a DDE are written as implicit functions of the parameters of interest, and the continuation equations are derived in the form of ordinary differential equations (ODEs). Numerical continuation is then employed to determine the characteristic roots at all points in a parametric space; the stability of the original DDE can then be easily determined. A key advantage of the proposed method is that a system of linearly independent ODEs is solved rather than the typical strategy of solving a large eigenvalue problem at each grid point in the domain. Thus, the CCR method significantly reduces the computational effort required to determine the stability of DDEs. As we demonstrate with several examples, the CCR method generates highly accurate stability charts, and does so up to 10 times faster than the Galerkin approximation method.