  In this contribution, we introduce numerical continuation methods and
bifurcation theory, techniques which find their roots in the study of dynamical
systems, to the problem of tracing the parameter dependence of bound and
resonant states of the quantum mechanical Schr\"odinger equation. We extend
previous work on the subject to systems of coupled equations. Bound and
resonant states of the Schr\"odinger equation can be determined through the
poles of the S-matrix, a quantity that can be derived from the asymptotic form
of the wave function. We introduce a regularization procedure that essentially
transforms the S-matrix into its inverse and improves its smoothness
properties, thus making it amenable to numerical continuation. This allows us
to automate the process of tracking bound and resonant states when parameters
in the Schr\"odinger equation are varied. We have applied this approach to a
number of model problems with satisfying results.
