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

TL;DR

This paper introduces a set-oriented numerical method for uncertainty quantification in dynamical systems with uncertain parameters, enabling analysis of long-term behavior without relying on polynomial chaos.

Contribution

It extends classical set-oriented methods to the UQ context, providing a global approach to analyze attractors and invariant measures under uncertainty.

Findings

Effective in approximating global attractors under uncertainty

Demonstrated efficiency through numerical examples

Does not depend on polynomial chaos techniques

Abstract

In this article, we develop a set-oriented numerical methodology which allows 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 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, and we illustrate the use and efficiency of these techniques by a couple of numerical examples.