Metric Properties of Attractors for Vector Fields via Bounded, Nonautonomous Control

Abstract

Attractors organize long-term behavior in dynamical systems, and their robustness against perturbations reflects both the resilience of a model system and the likelihood that the inexact model captures essential features of reality. Quantifying attractor strength requires metric information beyond the topological setting of classical dynamics. We extend the theory of “intensity of attraction,” first developed by McGehee for maps on compact metric spaces, to the context of continuous-time dynamics determined from a vector field on Euclidean space. Intensity measures the strength of transient dynamics within a domain of attraction, and can be computed by tracking the sets reachable from the attractor under bounded, nonautonomous control. We connect bounded control systems to multiflows, a nascent framework of multivalued dynamics. A connection between reachable sets and isolating blocks implies that an attractor’s intensity not only reflects its capacity to retain solutions under time-varying perturbations, but also gives a lower bound on the distance the attractor continues in the space of vector fields.