Informally, I am admitting as “stable” any dynamical system which does not explode super-polynomially fast; We can think of these as systems where if the system is not stationary then at least the rate of change might be.
Here I would like to think how to parameterize stable systems, and how to discover if systems are stable. This in a general context, which can be extremely hard in interesting systems. But often stability questions can be simpler in the context of linear systems, where is is about polynomial root-finding (perversely, polynomial root-finding is itself a famously chaotic system).
In a general setting should probably look at stuff like Lyapunov exponents to quantify stability.
Interesting connections here — we can also think about the relationships between stability and ergodicity, and criticality. Considering stability of neural networks turns out to produce some nice ideas.