  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.
