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Asymptotic dynamics of ordinary differential equations (ODEs) are commonlyunderstood by looking at eigenvalues of a matrix, and transient dynamics can bebounded above and below by considering the corresponding pseudospectra. Whileasymptotics for other classes of differential equations have been studied usingeigenvalues of a (nonlinear) matrix-valued function, there are no analogouspseudospectral bounds on transient growth. In this paper, we propose extensionsof the pseudospectral results for ODEs first to higher order ODEs and then todelay differential equations (DDEs) with constant delay. Results areillustrated with a discretized partial delay differential equation and a modelof a semiconductor laser with phase-conjugate feedback.

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