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

TL;DR

This paper introduces the concept of intensity of attraction, a metric-based measure of attractor robustness in ODEs, connecting control theory and Conley theory to quantify how much perturbation an attractor can withstand.

Contribution

It defines a new metric for attractor robustness called intensity of attraction, linking control-theoretic reachability with topological invariants in dynamical systems.

Findings

Intensity quantifies attractor robustness to vector field perturbations.

Main theorem connects intensity with the size of perturbations that preserve qualitative dynamics.

Provides a new tool for resilience analysis in ecological models.

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.