Computing Resilience Measures in Dynamical Systems

Abstract

Resilience broadly describes a quality of withstanding perturbations. Measures of system resilience have gathered increasing attention across applied disciplines, yet existing metrics often lack computational accessibility and generalizability. In this work, we review the literature on resilience measures through the lens of dynamical systems theory and numerical methods. In this context, we reformulate pertinent measures into a general form and introduce a resource-efficient algorithm designed for their parallel numerical estimation. By coupling these measures with a global continuation of attractors, we enable their consistent evaluation along system parameter changes. The resulting framework is modular and easily extendable, allowing for the incorporation of new resilience measures as they arise. We demonstrate the framework on a range of illustrative dynamical systems, revealing key differences in how resilience changes across systems. This approach provides a more global perspective compared to traditional linear stability metrics used in local bifurcation analysis, which can overlook inconspicuous but significant shifts in system resilience. This work opens the door to genuinely novel lines of inquiry, such as the development of new early warning signals for critical transitions or the discovery of universal scaling behaviours. All code and computational tools are provided as an open-source contribution to the DynamicalSystems.jl software library.