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

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.