Wilson Dan
Department of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, Tennessee 37996, USA.
Chaos. 2020 Jan;30(1):013121. doi: 10.1063/1.5126122.
Phase-amplitude reduction is of growing interest as a strategy for the reduction and analysis of oscillatory dynamical systems. Augmentation of the widely studied phase reduction with amplitude coordinates can be used to characterize transient behavior in directions transverse to a limit cycle to give a richer description of the dynamical behavior. Various definitions for amplitude coordinates have been suggested, but none are particularly well suited for implementation in experimental systems where output recordings are readily available but the underlying equations are typically unknown. In this work, a reduction framework is developed for inferring a phase-amplitude reduced model using only the observed model output from an arbitrarily high-dimensional system. This framework employs a proper orthogonal reduction strategy to identify important features of the transient decay of solutions to the limit cycle. These features are explicitly related to previously developed phase and isostable coordinates and used to define so-called data-driven phase and isostable coordinates that are valid in the entire basin of attraction of a limit cycle. The utility of this reduction strategy is illustrated in examples related to neural physiology and is used to implement an optimal control strategy that would otherwise be computationally intractable. The proposed data-driven phase and isostable coordinate system and associated reduced modeling framework represent a useful tool for the study of nonlinear dynamical systems in situations where the underlying dynamical equations are unknown and in particularly high-dimensional or complicated numerical systems for which standard phase-amplitude reduction techniques are not computationally feasible.
相位-振幅约简作为一种用于约简和分析振荡动力系统的策略，正日益受到关注。用振幅坐标增强广泛研究的相位约简，可用于刻画与极限环横向方向上的瞬态行为，从而更丰富地描述动力行为。人们已提出多种振幅坐标的定义，但在实验系统中，这些定义都不太适合实施，因为在实验系统中，输出记录很容易获取，但通常未知其基础方程。在这项工作中，我们开发了一个约简框架，用于仅从任意高维系统的观测模型输出推断相位-振幅约简模型。该框架采用适当的正交约简策略来识别极限环解的瞬态衰减的重要特征。这些特征与先前开发的相位坐标和等稳坐标明确相关，并用于定义在极限环的整个吸引域内有效的所谓数据驱动的相位坐标和等稳坐标。这种约简策略的实用性在与神经生理学相关的例子中得到了说明，并用于实施一种否则在计算上难以处理的最优控制策略。所提出的数据驱动的相位坐标和等稳坐标系以及相关的约简建模框架，是研究非线性动力系统的有用工具，适用于基础动力学方程未知的情况，特别是对于标准相位-振幅约简技术在计算上不可行的高维或复杂数值系统。
