利用不确定性量化在非线性振荡系统中定位孤立岛
Locating isolas in nonlinear oscillator systems using uncertainty quantification
AI总结：
本文提出一种通用的不确定性量化框架，用于分析具有比例参数不确定性非线性动力系统中的分岔行为，通过结合持续方法和参数灵敏度及极值条件，有效捕捉不确定性引起的分岔边界和响应边界的位移。
摘要：
非线性动力系统中的参数不确定性可以根本改变分岔行为，导致响应质量的变化。预测在这样的不确定性下的运行边疆/包络是关键但具有挑战性：传统不确定性量化（UQ）方法在分岔边界上难以高效传播不确定性，因为响应梯度变得奇异，解分支出现/消失。本文提出一种通用的UQ框架，用于分析具有比例参数不确定性的非线性动力系统中的分岔行为，该方法系统地结合了持续方法、参数灵敏度和极值条件。该方法采用两步方案：首先，当不确定性域从确定性参考点扩展时，极值响应点的轨迹被追踪；其次，这些极值点被追踪，当分岔参数变化时，从而确定整个过程中的最大和最小响应边疆。持续问题的规模与不确定参数的数量成线性关系，从而实现高效的分析。该方法在具有多种分岔现象的二自由度非线性振荡器上得到验证，包括多解、模态相互作用和对称破坏。在所有情况下，该框架都有效地捕捉了不确定性引起的分岔边界和响应边疆的位移。值得注意的是，该方法揭示了参数不确定性在分岔结构中引起拓扑变化，包括孤立响应分支的出现，而这些在确定性系统中使用参考参数时是不存在的。
英文摘要：
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.
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