A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty

Abstract

In this article, we develop a set-oriented numerical methodology which allows us to perform uncertainty quantification (UQ) for dynamical systems from a global point of view. That is, for systems with uncertain parameters we approximate the corresponding global attractors and invariant measures in the related stochastic setting. Our methods do not rely on generalized polynomial chaos techniques. Rather, we extend classical set-oriented methods designed for deterministic dynamical systems [M. Dellnitz and A. Hohmann, Numer. Math., 75 (1997), pp. 293--317; M. Dellnitz and O. Junge, SIAM J. Numer. Anal., 36 (1999), pp. 491--515] to the UQ-context, and this allows us to analyze the long-term uncertainty propagation. The algorithms have been integrated into the software package GAIO [M. Dellnitz, G. Froyland, and O. Junge, Ergodic Theory, Analysis, and Efficient Simulation of Dynamical Systems, Springer, Berlin, 2001, pp. 145--174], and we illustrate the use and efficiency of these techniques with a couple of numerical examples.