Dynamics of the boundary map of a system with spherical noise

Random dynamical systems with bounded noise are studied. In such systems, all trajectories are typically attracted to minimal sets, which are attractors. The problem of directly determining a minimal set is nontrivial, because one has to deal with a poorly investigated object, namely, with a set-valued map. However, there is an approach that allows reducing this problem to finding the invariant set of an ordinary discrete dynamical system, namely, of a boundary map. The minimal invariant sets are considered for the class of random dynamical systems consisting of invertible linear maps with bounded spherical noise. An exhaustive description of boundary maps is given in the case of typical linear contraction. It is found that the boundary map is then a Morse–Smale diffeomorphism whose global attractor uniquely determines the boundary of the minimal set of a random system.