As a confused undergrad first learning about random dynamical systems on my own (outside of any classes), this really bothered me. How could the Markov assumpti

As a confused undergrad first learning about random dynamical systems on my own (outside of any classes), this really bothered me. How could the Markov assumption be so powerful when it's clearly trivial to add memory and retain the Markov property?! 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.