强度——一种量化常微分方程中吸引子鲁棒性的度量方法
Intensity -- A Metric Approach to Quantifying Attractor Robustness in ODEs
AI总结：
本文提出“吸引强度”这一控制论度量，用以量化 ODE 吸引子对时变与时不变向量场扰动的鲁棒性，并可用于生态恢复力评估。
摘要：
尽管数学模型与现实并不完全吻合，但动力学对象对扰动的鲁棒性有助于架起从理论动力学到现实动力学系统的桥梁。结构稳定性和孤立不变集的经典理论处理的是定性动力学对足够小误差的鲁棒性。然而，它们并不能说明在系统的定性行为发生根本性变化之前，扰动究竟可以达到多大。在此，我们引入一个量，即吸引强度，它以度量方式衡量吸引子的鲁棒性。在 $\mathbb{R}^n$ 上的常微分方程背景下，我们考虑对时间相关或时间无关的向量场扰动的鲁棒性。我们在控制论框架中定义强度，其基础是将轨迹引导出吸引域所需的控制量大小。我们的主要结果是，强度也量化了吸引子对时间无关向量场扰动的鲁棒性；我们通过将控制论中的可达集与 Conley 理论中的隔离块联系起来证明了这一点。除了在一个新的度量框架中处理经典的鲁棒性问题外，吸引强度还为生态学应用中的恢复力量化提供了一种新颖工具。与许多恢复力度量不同，强度能够检测吸引域内瞬态动力学的强度。
英文摘要：
Although mathematical models do not fully match reality, robustness of dynamical objects to perturbation helps bridge from theoretical to real-world dynamical systems. Classical theories of structural stability and isolated invariant sets treat robustness of qualitative dynamics to sufficiently small errors. But they do not indicate just how large a perturbation can become before the qualitative behavior of our system changes fundamentally. Here we introduce a quantity, intensity of attraction, that measures the robustness of attractors in metric terms. Working in the setting of ordinary differential equations on $\mathbb{R}^n$, we consider robustness to vector field perturbations that are time-dependent or -independent. We define intensity in a control-theoretic framework, based on the magnitude of control needed to steer trajectories out of a domain of attraction. Our main result is that intensity also quantifies the robustness of an attractor to time-independent vector field perturbations; we prove this by connecting the reachable sets of control theory to isolating blocks of Conley theory. In addition to treating classical questions of robustness in a new metric framework, intensity of attraction offers a novel tool for resilience quantification in ecological applications. Unlike many measurements of resilience, intensity detects the strength of transient dynamics in a domain of attraction.
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