Pseudospectral bounds on transient growth for higher order and constant delay differential equations

TL;DR

This paper extends pseudospectral bounds to analyze transient growth in higher order and delay differential equations, providing new tools for understanding their dynamics beyond eigenvalue analysis.

Contribution

It introduces pseudospectral bounds for higher order ODEs and delay differential equations, filling a gap in transient growth analysis for these classes.

Findings

Pseudospectral bounds effectively estimate transient growth in higher order ODEs.

The approach is applicable to delay differential equations with constant delay.

Illustrations include a discretized partial delay differential equation and a semiconductor laser model.

Abstract

Asymptotic dynamics of ordinary differential equations (ODEs) are commonly understood by looking at eigenvalues of a matrix, and transient dynamics can be bounded above and below by considering the corresponding pseudospectra. While asymptotics for other classes of differential equations have been studied using eigenvalues of a (nonlinear) matrix-valued function, there are no analogous pseudospectral bounds on transient growth. In this paper, we propose extensions of the pseudospectral results for ODEs first to higher order ODEs and then to delay differential equations (DDEs) with constant delay. Results are illustrated with a discretized partial delay differential equation and a model of a semiconductor laser with phase-conjugate feedback.