Computation of uncertainty distributions in complex dynamical systems

The computation of the stationary distribution of an uncertain nonlinear dynamical system is an important tool in analysis of the long term behavior of the system. One common approach is to use a Monte Carlo type method. However, that type of method requires many simulations runs to achieve a reasonable accuracy and can be computationally excessive. In this paper we formulate an alternative approach based on the theory of random dynamical systems to solve this problem. Using the properties of the invariant measure of the Perron-Frobenius operator for the dynamical systems we obtain a simple characterization of the stationary distribution. The state space is discretized to obtain a finite dimensional approximation for the infinite dimensional Perron-Frobenius operator. Furthermore, an efficient subdivision algorithm for state space partition is discussed. The approach is demonstrated through a catalytic reactor system.