Invariant manifolds are fundamental geometric structures in the field of nonlinear dynamical systems, providing insights into the system's long-term global behavior. In the context of nondeterministic dynamical systems (e.g., stochastic or random dynamical systems), the concept of invariant manifolds generalizes but becomes more nuanced due to the presence of randomness or uncertainty and the definition of invariance must account for the probabilistic behavior. To address this concept, necessary definitions from the measurable dynamics' theory are introduced, including the flow map, the transfer operator and its dual in both closed and open spaces, and the classical Ulam discretization over the space of constant distributions along with its dual. Next, invariant manifolds for nondeterministic systems are defined, and a proof of existence is provided for a marginal distribution fWu over the unstable manifold, along with its dual observable gWs over the stable manifold. Finally, a discretization strategy for the open-flow transfer operator is presented, along with a method for computing fWu and gWs. In summary, the theory of measurable dynamics is employed to define and prove the existence of unstable manifold distributions and stable manifold observables via the spectrum of an open-flow transfer operator. In addition, a computational discretization procedure, based on the Ulam method, is introduced. The results of the electrically actuated microarch reveal three interconnected stochastic phenomena: diminished convergence probability, expansion of the stable manifold across its basin, and fusion of the unstable manifold with the attractor.
Benedetti et al. (Sun,) studied this question.