Recently the effect of a quiescent phase (or dormant/resting phase in applications) on
the dynamics of a system of differential equations has been investigated, in particular with respect
to stability properties of stationary points. It has been shown that there is a general phenomenon
of stabilization against oscillations which can be cast in rigorous form. Here we investigate, for
homogeneous systems, the effect of a quiescent phase, and more generally, a phase with slower
dynamics. We show that each exponential solution of the original system produces two exponential
solutions of the extended system whereby the stability properties can be controlled.