My PhD thesis was about the extension of smooth functions. In the 1960s, ideas in this area were studied under the umbrella of ideals of smooth functions, and had applications to catastrophe theory. So I spent some time reading about this topic. If you want a reading recommendation, the book "Differentiable Germs and Catastrophes" I remember being good, as well as "Stable Mappings and their Singularities".

I bring this up because I believe that at the time catastrophe theory was seen as a "widely-applicable science of systems". Or at least some practitioners tried to sell it as such. This point of view eventually soured to the detriment of catastrophe theory, which cleared out. I don't think this was a good thing: catastrophe theory (the study of the singularities of smooth maps and their consequences to dynamical systems) is an interesting topic with many remaining open questions. But it was seen as cringe that people were, e.g., using Whitney's classification of the generic singularities of planar maps to try to say something about predator/prey dynamics or whatever. Any claim about applications of catastrophe theory was infected with this stink, and so people lost interest in it.