What time varying-parameters do to a model like this is to make the long-term equilibrium to stop being a specific point in the phase diagram. The zero-change loci shift and move around, and so does their intersection.
A simple thing that you could do, at least from a pedagogical point of view, is to reproduce the effect of "structural breaks" and not smooth time-variation. So for a certain time period, a parameter of the model was fixed at a certain level, but then it jumped to a different level (say depreciation changed because in the past it was more buildings and machinery, now it is more IT, software and intangibles).
Such an abrupt change essentially discretizes a smooth time-variation, and it is an acceptable approximation to it.
This means that one draws two pairs of zero-change loci in the phase diagram, one representing the situation before and one after the structural break. And one shows how the economy behaves by jumping from the one saddle-path to the other.
See this blog post of mine where I implemented this approach to reflect in simple descriptive terms the current depression of the Greek Economy.