AI 中文总结
研究将柯普曼 - 希尔方法从ODEs推广到DAEs以分析其周期解稳定性，核心是建立高维线性时不变DAE控制扰动动力学，通过两个学术机械系统验证方法适用性及对DAE指标的不敏感性。
AI 中文摘要
微分代数方程（DAEs）和常微分方程（ODEs）的周期解可通过谐波平衡法（HBM）确定，该方法是一种频域方法，用截断傅里叶级数逼近解。本文将最初为ODEs开发的用于确定HBM找到的周期解稳定性的柯普曼 - 希尔方法推广到DAEs。对于DAEs的柯普曼 - 希尔方法的核心思想是建立一个线性时不变但高维的DAE来近似控制周期解周围小允许扰动的动力学。与ODE情况的关键区别在于，这个线性时不变DAE的演化不是简单地由矩阵指数给出，而是由涉及Drazin逆的更复杂表达式给出，使得所得的单值矩阵奇异。不过，即使在DAE情况下，单值矩阵和希尔矩阵之间的这种新关系基本上由一个单一公式给出，这是本文的主要结果。两个学术机械系统，一个表述为指标 - 3 DAE的数学摆和一个具有切换指标的非光滑摩擦双质量振荡器，证明了所提出方法的适用性及其对DAE指标的不敏感性。
英文摘要
Periodic solutions of differential-algebraic equations (DAEs) and ordinary differential equations (ODEs) can be determined using the harmonic balance method (HBM), which is a frequency-domain approach that approximates the solution by its truncated Fourier series. The Koopman-Hill method, a method to determine the stability of periodic solutions found by HBM, and originally developed for ODEs, is generalized to DAEs in this work. Analogously to the ODE case, the core idea of the proposed Koopman-Hill method for DAEs is to establish a linear time-invariant but high-dimensional DAE which approximately governs the dynamics of small admissible perturbations around the periodic solution. The crucial difference to the ODE case is the fact that the evolution of this linear time-invariant DAE is not simply given by a matrix exponential, but by a more complicated expression involving a Drazin inverse, rendering the resulting monodromy matrix singular. Still, even in the DAE case, this novel relationship between the monodromy matrix and the Hill matrix is essentially given by one single formula, which is the main result of this work. Two academic mechanical systems, a mathematical pendulum formulated as an index-3 DAE and a nonsmooth frictional two-mass oscillator with switching index, demonstrate the applicability of the proposed method and its blindness to the DAE's index.
