Preface: This question is now asking about dynamical stability in a particular sense, and whether its existence can be inferred from data. It is motivated by commentary below the question "Is there a statistical analogue of the quantum mechanical tradeoff between information gain and system disturbance?".
Lyapunov stability is a property of certain autonomous dynamical systems. It is the sort of property that is sometimes provable when given the rate equations of a system using $\epsilon,\delta$ proofs. But it isn't obvious how, or even if, such a notion can be translated into a statistical estimator.
Has someone developed an estimator that indicates (or quantifies) Lyapunov-motivated stability of a system?
