The importance of considering the presence of noise and uncertainty has become increasingly evident in real-world applications during the last few decades. With an assumption of bounded noise, the stationary distributions of the corresponding random dynamical systems are typically non-unique and supported on compact sets, allowing for a topological characterisation of the dynamical situation. The collection of trajectories with all possible noise realisations can be described at the topological level as a deterministic set-valued dynamical system. We are interested in the bifurcation of the stationary distribution of the random dynamical system, which turns out to be invariant sets of the set-valued system.
Set-valued dynamical systems are notoriously challenging to analyse both theoretically and computationally, as they are defined on the set of compact subsets, which is infinite-dimensional and not amenable to analysis techniques typically used to understand bifurcation problems. We introduce a single-valued, finite-dimensional boundary map, inspired by the normal bundle of the boundary of invariant sets. We present an example of Hénon map with bounded noise where the bifurcation of the stationary distribution can be detected by traditional bifurcation of the boundary map. We also show some theoretical results on linear map and persistence of the normal bundle of invariant set’s boundary.