Critical Delays and Polynomial Eigenvalue Problems

TL;DR

This paper introduces a novel method for computing critical delays in delay differential equations by solving quadratic and polynomial eigenvalue problems, enabling the characterization of purely imaginary eigenvalues and critical surfaces.

Contribution

The work presents a new eigenvalue problem-based approach to parameterize critical delays, including a closed-form solution for scalar cases and handling of commensurate delays.

Findings

Method effectively computes critical delays for DDEs.

Provides visualizations and examples demonstrating the approach.

Establishes connections with existing methods through eigenvalue problem formulations.

Abstract

In this work we present a new method to compute the delays of delay differential equations (DDEs), such that the DDE has a purely imaginary eigenvalue. For delay differential equations with multiple delays, the critical curves or critical surfaces in delay space (that is, the set of delays where the DDE has a purely imaginary eigenvalue) are parameterized. We show how the method is related to other works in the field by treating the case where the delays are integer multiples of some delay value, i.e., commensurate delays. The parametrization is done by solving a {\em quadratic eigenvalue problem} which is constructed from the vectorization of a matrix equation and hence typically of large size. For commensurate delay differential equations, the corresponding equation is a polynomial eigenvalue problem. As a special case of the proposed method, we find a closed form for a parameterization of the critical surface for the scalar case. We provide several examples with visualizations where the computation is done with some exploitation of the structure of eigenvalue problems.