Uncertainty Analysis of Limit Cycle Oscillations in Nonlinear Dynamical Systems with the Fourier Generalized Polynomial Chaos Expansion

TL;DR

This paper introduces a Fourier generalized Polynomial Chaos expansion method to analyze and predict uncertainties in limit cycle oscillations of nonlinear dynamical systems, demonstrated on engineering and biological models.

Contribution

It presents a novel spectral approach combining Fourier analysis with polynomial chaos for uncertainty quantification in nonlinear oscillatory systems.

Findings

Accurately predicts distributions of limit cycle oscillations.

Provides uncertainty estimates for base frequency in self-excited systems.

Shows potential for adaptive approximation using coefficient sparsity.

Abstract

In engineering, simulations play a vital role in predicting the behavior of a nonlinear dynamical system. In order to enhance the reliability of predictions, it is essential to incorporate the inherent uncertainties that are present in all real-world systems. Consequently, stochastic predictions are of significant importance, particularly during design or reliability analysis. In this work, we concentrate on the stochastic prediction of limit cycle oscillations, which typically occur in nonlinear dynamical systems and are of great technical importance. To address uncertainties in the limit cycle oscillations, we rely on the recently proposed Fourier generalized Polynomial Chaos expansion (FgPC), which combines Fourier analysis with spectral stochastic methods. In this paper, we demonstrate that valuable insights into the dynamics and their variability can be gained with a FgPC analysis, considering different benchmarks. These are the well-known forced Duffing oscillator and a more complex model from cell biology in which highly non-linear electrophysiological processes are closely linked to diffusive processes. With our spectral method, we are able to predict complicated marginal distributions of the limit cycle oscillations and, additionally, for self-excited systems, the uncertainty in the base frequency. Finally we study the sparsity of the FgPC coefficients as a basis for adaptive approximation.