Locating isolas in nonlinear oscillator systems using uncertainty quantification

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.

Figures (10)

• Figure 1: Uncertainty quantification for the Duffing oscillator. The forced response curve of the reference system (without uncertainty) is shown as a black line in the projection of excitation frequency against displacement amplitude. The response margins of the uncertain system are obtained via a simulation-based UQ method. (Online version in colour.)
• Figure 2: Schematic diagram of the proposed uncertainty quantification method. $(a)$ Uncertainty expansion step: the extremal solutions are traced as the uncertainty domain expands from null to the marginal size. $(b)$ Uncertainty propagation step: the metric margins are traced as the uncertainty propagates through the bifurcation parameter to the extremal solutions. (Online version in colour.)
• Figure 3: Uncertainty quantification for a two-mode system with sufficiently separated eigenfrequencies. $(a)$ Forced response curve for the reference system (without uncertainty) in the projection of excitation frequency against displacement amplitude of the first mass. $(b)$, $(c)$, and $(d)$ Metric margins and margin-governing uncertainties for the uncertain system with an uncertainty level of $R = 0.1$. (Online version in colour.)
• Figure 4: Uncertainty quantification for a two-mode system featuring modal interactions. $(a)$ Forced response curve for the reference system in the projection of excitation frequency against displacement amplitude of the second mass. The embedded plots $(i)$, $(ii)$, and $(iii)$ show the periodic orbits for near-resonant POs in the configuration space. $(b)$, $(c)$, and $(d)$ Metric margins and margin-governing uncertainties for the uncertain system with an uncertainty level of $R = 0.1$. (Online version in colour.)
• Figure 5: Sensitivity of the internally resonant region to the level of uncertainty. Metric margins for the uncertain system in the projection of excitation frequency against displacement amplitude of the second mass. Four uncertainty levels of $R = 0.025,~0.050,~0.075,~0.100$ are shown as thick lines; whilst the reference metric is shown as a thin line. (Online version in colour.)