标题： 一种径向和切向框架用于研究瞬态反应性 显示英文标题

标题： A Radial and Tangential Framework for Studying Transient Reactivity

摘要： 在建模生物系统和其他应用中，一个重要的反复出现的问题是，这些系统不仅在动态吸引子上保持健康状态，而且在远离这些吸引子的瞬态偏离期间是否也能保持健康状态。 对于常微分方程模型，这些偏离不仅仅是由非线性引起的。即使在一个具有原点全局吸引子的线性自治系统中，一些轨迹也会暂时远离原点，最终被渐近吸引。 Reactivity（反应性）由Neubert和Caswell于1997年定义，它通过测量径向增长的最大瞬时速率来捕捉扰动瞬态放大的概念。 在本文中，我们引入了一个新的框架，用于分析二维线性常微分方程中的反应性和瞬态动力学，该框架使用了矢量场的径向和切向分解。 我们建立了如何通过这种视角来看待系统的特征结构，并引入了一种正交矢量和正交值的匹配结构，这些结构描述了径向速度变为正的区域。 从这个角度出发，我们获得了关于状态空间中出现正径向增长的区域以及解轨迹如何穿过这些区域的几何洞察。 此外，我们提出了四种标准矩阵形式，这些形式可以表征瞬态和渐近行为，并且更直接地突出反应性特征。 最后，我们将我们的框架应用于探索反应性的极限和最大放大，并描述瞬态反应性如何在非自治线性系统中累积，从而导致渐近不稳定的behavior。 显示英文摘要

摘要： In modeling biological systems and other applications, an important recurring question is whether those systems maintain healthy regimes not just at dynamic attractors but also during transient excursions away from those attractors. For ODE models, these excursions are not due only to nonlinearities. Even in a linear, autonomous system with a global attractor at the origin, some trajectories will move transiently away from the origin before eventually being attracted asymptotically. Reactivity, defined by Neubert and Caswell in 1997, captures this idea of transient amplification of perturbations by measuring the maximum instantaneous rate of radial growth. In this paper, we introduce a novel framework for analyzing reactivity and transient dynamics in two-dimensional linear ODEs using a radial and tangential decomposition of the vector field. We establish how to view the eigen-structure of the system through this lens and introduce a matching structure of orthovectors and orthovalues that characterize where the radial velocity becomes positive. From this perspective, we gain geometric insight about the regions of state space where positive radial growth occurs and how solutions trajectories traverse through these regions. Additionally, we propose four standard matrix forms that characterize both transient and asymptotic behavior and which highlight reactivity features more directly. Finally, we apply our framework to explore the limits of reactivity and maximal amplification and to characterize how transient reactivity can accumulate in nonautonomous linear systems to result in asymptotically unstable behavior.