# How to find simple nonlocal stability and resilience measures

## TL;DR

This paper introduces simple, nonlocal stability and resilience measures for dynamical systems that account for large perturbations, overcoming limitations of traditional local linearization methods, and demonstrates their effectiveness on various models.

## Contribution

The paper proposes easy-to-implement nonlocal measures for stability and resilience based on basin estimates and return times, filling a gap in existing methods.

## Key findings

- Measures detect dynamic behaviors missed by local stability.
- Approach is applicable to diverse models including economic, mechanical, and biological systems.
- Methods are computationally simple and accessible on standard computers.

## Abstract

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 can not 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 as well as on a stage-structured population model, and conclude that the suggested measures detect dynamic behaviour, 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 towards 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.