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

TL;DR

This work tackles the challenge of uncertainty in limit cycle oscillations of nonlinear dynamical systems by introducing the Fourier generalized Polynomial Chaos expansion (FgPC), which couples Harmonic Balance with intrusive generalized Polynomial Chaos to build a time- and parameter-dependent surrogate $x_{HN}(t,θ)$. A deflation technique is integrated to uncover multiple LCO solutions under the same parameter set. The method is demonstrated on the Duffing oscillator with uncertain linear stiffness and on a pancreatic β-cell electrophysiology model with uncertain ATP, achieving accurate marginal distributions and base-frequency statistics while offering large computational savings over Monte Carlo. Limitations arise near bifurcations with discontinuities where the HB-based approach struggles, but overall the framework provides a scalable, sparse representation for uncertainty quantification in self-excited and forced nonlinear oscillators.

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.

Figures (16)

• Figure 1: Visual representation of Fourier generalized Polynomial Chaos expansion
• Figure 2: Backbone curve of the Duffing oscillator for different values of $\alpha$
• Figure 3: Magnitude of each coefficient for every combination of the harmonic $H$ and polynomial degree $N$ for poition $x$ of the Duffing oscillator
• Figure 4: Relative error of the FgPC method for every combination between the harmonic $H$ and polynomial degree $N$ for the Duffing oscillator
• Figure 5: Large stable FgPC solution of the Duffing oscillator with $H=5$ and $N=12$