  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.
