Wilson Dan
Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA.
Phys Rev E. 2021 Feb;103(2-1):022211. doi: 10.1103/PhysRevE.103.022211.
Isostable coordinates provide a convenient framework for understanding the transient behavior of dynamical systems with stable attractors. These isostable coordinates are often used to characterize the slowest decaying eigenfunctions of the Koopman operator; by neglecting the rapidly decaying Koopman eigenfunctions a reduced order model can be obtained. Existing work has focused primarily on nondegenerate isostable coordinates, that is, isostable coordinates that are associated with eigenvalues that have identical algebraic and geometric multiplicities. Current isostable reduction methods cannot be applied to characterize the decay associated with a defective eigenvalue. In this work, a degenerate isostable framework is proposed for use when eigenvalues are defective. These degenerate isostable coordinates are investigated in the context of various reduced order modeling frameworks that retain many of the important properties of standard (nondegenerate) isostable reduced modeling strategies. Reduced order modeling examples that require the use of degenerate isostable coordinates are presented with relevance to both circadian physiology and nonlinear fluid flows.
等稳态坐标为理解具有稳定吸引子的动力系统的瞬态行为提供了一个便利的框架。这些等稳态坐标常被用于刻画柯普曼算子最慢衰减的本征函数；通过忽略快速衰减的柯普曼本征函数，可以得到一个降阶模型。现有工作主要集中在非退化等稳态坐标上，即与具有相同代数重数和几何重数的特征值相关联的等稳态坐标。当前的等稳态约化方法不能用于刻画与亏损特征值相关的衰减。在这项工作中，当特征值为亏损时，提出了一个退化等稳态框架。在各种保留了标准（非退化）等稳态约化建模策略许多重要性质的降阶建模框架的背景下，对这些退化等稳态坐标进行了研究。给出了需要使用退化等稳态坐标的降阶建模示例，这些示例与昼夜生理和非线性流体流动都相关。
