Locating isolas in nonlinear oscillator systems using uncertainty quantification

TL;DR

This paper introduces a novel uncertainty quantification framework that efficiently locates isolas and predicts bifurcation boundary shifts in nonlinear oscillators with parametric uncertainty, aiding robust system design.

Contribution

The paper presents a systematic UQ method combining continuation, sensitivities, and extremal conditions to analyze bifurcations under uncertainty, including the detection of isolas.

Findings

Efficiently captures shifts in bifurcation boundaries due to uncertainty.

Reveals emergence of isolated response branches (isolas) caused by parametric uncertainty.

Demonstrates applicability on a nonlinear oscillator with complex bifurcation phenomena.

Abstract

Parametric uncertainty in nonlinear dynamical systems can fundamentally alter bifurcation behaviour, leading to qualitative response changes. Predicting operating margins/envelopes under such uncertainties is critical but challenging: conventional uncertainty quantification (UQ) methods struggle to efficiently propagate uncertainties across bifurcation boundaries, where response gradients become singular and solution branches emerge/vanish. We present a general UQ framework for bifurcation analysis of nonlinear dynamical systems with proportional parametric uncertainty, which systematically integrates continuation methods with parametric sensitivities and extremal conditions. The approach uses a two-step scheme: first, the loci of extremal response points are traced as the uncertainty domain is expanded from a deterministic reference point; second, these extremal points are tracked as the bifurcation parameter varies, thus determining the maximum and minimum response margins throughout. The continuation problem scales linearly with the number of uncertain parameters, enabling efficient analysis. The method is demonstrated on a two-degree-of-freedom nonlinear oscillator exhibiting a range of bifurcation phenomena, including multiple solutions, modal interactions, and symmetry breaking. In all cases, the framework efficiently captures uncertainty-induced shifts in bifurcation boundaries and response margins. Notably, the method reveals that parametric uncertainty induces topological changes in the bifurcation structure, including the emergence of an isolated response branch that is absent in the deterministic system with the reference parameters.