# Computing invariant sets of random differential equations using polynomial chaos

## TL;DR

This paper introduces a polynomial chaos-based method to efficiently compute and visualize invariant sets of random differential equations, enabling uncertainty quantification and faster sampling in complex dynamical systems.

## Contribution

It presents a novel approach using polynomial chaos to analyze invariant sets of RODEs, enhancing computational efficiency and visualization capabilities.

## Key findings

- Efficient computation of invariant sets for RODEs using PC.
- Successful application to predator-prey and Lorenz systems.
- Demonstrated numerical efficiency with adaptive PC methods.

## Abstract

Differential equations with random parameters have gained significant prominence in recent years due to their importance in mathematical modelling and data assimilation. In many cases, random ordinary differential equations (RODEs) are studied by using Monte-Carlo methods or by direct numerical simulation techniques using polynomial chaos (PC), i.e., by a series expansion of the random parameters in combination with forward integration. Here we take a dynamical systems viewpoint and focus on the invariant sets of differential equations such as steady states, stable/unstable manifolds, periodic orbits, and heteroclinic orbits. We employ PC to compute representations of all these different types of invariant sets for RODEs. This allows us to obtain fast sampling, geometric visualization of distributional properties of invariants sets, and uncertainty quantification of dynamical output such as periods or locations of orbits. We apply our techniques to a predator-prey model, where we compute steady states and stable/unstable manifolds. We also include several benchmarks to illustrate the numerical efficiency of adaptively chosen PC depending upon the random input. Then we employ the methods for the Lorenz system, obtaining computational PC representations of periodic orbits, stable/unstable manifolds and heteroclinic orbits.