SIAM Journal on Applied Dynamical Systems, Volume 19, Issue 1, Page 329-351, January 2020.
This paper addresses structures of state space in quasiperiodically forced dynamical systems. We develop a theory of ergodic partition of state space in a class of measure-preserving and dissipative flows, which is a natural extension of the existing theory for measure-preserving maps. The ergodic partition result is based on eigenspace at eigenvalue 0 of the associated Koopman operator, which is realized via time-averages of observables, and provides a constructive way to visualize a low-dimensional slice through a high-dimensional invariant set. We apply the result to the systems with a finite number of attractors and show that the time-average of a continuous observable is well defined and reveals the invariant sets, namely, a finite number of basins of attraction. We provide a characterization of invariant sets in the quasiperiodically forced systems. A theoretical result on uniform boundedness of the invariant sets is presented. The series of theoretical results enables numerical analysis of invariant sets in the quasiperiodically forced systems based on the ergodic partition and time-averages. Using this, we analyze a nonlinear model of complex power grids that represents the short-term swing instability, named the coherent swing instability. We show that our theoretical results can be used to understand stability regions in such complex systems.

中文翻译：

---

准周期动力系统中的不变集

SIAM应用动力系统杂志，第19卷，第1期，第329-351页，2020年1月。
本文讨论准周期动力系统中的状态空间结构。我们在保存度量和耗散流的一类中发展了状态空间的遍历划分理论，这是对保存度量地图的现有理论的自然扩展。遍历分割结果基于相关联的Koopman算子的特征值0处的特征空间，该特征空间是通过可观察对象的时间平均值实现的，并提供了一种通过高维不变集可视化低维切片的构造方法。我们将结果应用于具有有限数量吸引子的系统，并表明连续可观测时间的均值定义明确，并揭示了不变集，即有限数量的吸引盆地。我们提供了准周期性系统中不变集的特征。给出了关于不变集的一致有界性的理论结果。一系列的理论结果使得能够基于遍历划分和时间平均对准周期性系统中的不变集进行数值分析。使用此方法，我们分析了表示短期摆动不稳定性的复杂电网非线性模型，称为相干摆动不稳定性。我们证明了我们的理论结果可用于理解这种复杂系统中的稳定区域。使用此方法，我们分析了表示短期摆动不稳定性的复杂电网非线性模型，称为相干摆动不稳定性。我们证明了我们的理论结果可用于理解这种复杂系统中的稳定区域。使用此方法，我们分析了表示短期摆动不稳定性的复杂电网非线性模型，称为相干摆动不稳定性。我们证明了我们的理论结果可用于理解这种复杂系统中的稳定区域。