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arXiv · 2012.10786

Intensity -- A Metric Approach to Quantifying Attractor Robustness in ODEs

Katherine J. Meyer·Richard P. McGehee
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Abstract

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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BibTeXRIS

Katherine J. Meyer, Richard P. McGehee. 2020-12-19. Intensity -- A Metric Approach to Quantifying Attractor Robustness in ODEs. https://doi.org/10.1137/20m138689x

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