How to find simple nonlocal stability and resilience measures

• Published in: Nonlinear dynamics Vol. 93; no. 2; pp. 887 - 908
• Main Author: Lundström, Niklas L. P.
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.07.2018; Springer Nature B.V
• Subjects: Attractors (mathematics); Automotive Engineering; Basin stability; Bistability; Classical Mechanics; Control; Dynamic stability; Dynamical Systems; Economic development; Economic models; Engineering; Global stability; Initial conditions; Mathematical models; Measure of resilience; Measure of stability; Mechanical Engineering; Mechanical systems; Nonlinear dynamics; Nonlinear systems; Original Paper; Perturbation; Random testing; Recovery rate; Resilience; Returntime; Sensitivity; Systems stability; Vibration; Random testing; Sensitivity; Initial conditions; Global stability; Measure of resilience; Returntime; Recovery rate; Measure of stability; Basin stability; Bistability
• ISSN: 0924-090X; 1573-269X; 1573-269X
• DOI: 10.1007/s11071-018-4234-x

Stability of dynamical systems is a central topic with applications in widespread areas such as economy, biology, physics and mechanical engineering. The dynamics of nonlinear systems may completely change due to perturbations forcing the solution to jump from a safe state into another, possibly dangerous, attractor. Such phenomena cannot be traced by the widespread local stability and resilience measures, based on linearizations, accounting only for arbitrary small perturbations. Using numerical estimates of the size and shape of the basin of attraction, as well as the systems returntime to the attractor after given a perturbation, we construct simple nonlocal stability and resilience measures that record a systems ability to tackle both large and small perturbations. We demonstrate our approach on the Solow–Swan model of economic growth, an electro-mechanical system, a stage-structured population model as well as on a high-dimensional system, and conclude that the suggested measures detect dynamic behavior, crucial for a systems stability and resilience, which can be completely missed by local measures. The presented measures are also easy to implement on a standard laptop computer. We believe that our approach will constitute an important step toward filling a current gap in the literature by putting forward and explaining simple ideas and methods, and by delivering explicit constructions of several promising nonlocal stability and resilience measures.