After reading through theorems and proofs for dynamical systems theory, and especially considering applications of that theory, I came up with basically two practical take-aways:
1. Complexity matters and the curse of dimensionality is real.
2. All those nice theorems (stability, convergence, splitting, invariance, bifurcations, ...) are stated with respect to a state space and make assumptions about the dynamics on that state space. So if you code up a state space representation that is history-aware with respect to the dynamics you really care about, then:
a) you might lose some theorems that hold for the original process, and
b) you sometimes realize the theorems lose a lot of power when they're telling you something about a tuple of things you care about instead of the actual thing you care about.
If I ever teach people about random dynamical systems, this explanation will come immediately after the definition of a Markov process and I'll continually remind my audience throughout the course. Understanding what goes wrong when a system is "technically" Markovian is probably the best way to obtain a deep intuition. Or at least, it was very helpful to me.