Framework for global stability analysis of dynamical systems.

Abstract

Dynamical systems that are used to model power grids, the brain, and other physical systems can exhibit coexisting stable states known as attractors. A powerful tool to understand such systems, as well as to better predict when they may "tip" from one stable state to the other, is global stability analysis. It involves identifying the initial conditions that converge to each attractor, known as the basins of attraction, measuring the relative volume of these basins in state space, and quantifying how these fractions change as a system parameter evolves. By improving existing approaches, we present a comprehensive framework that allows for global stability analysis of dynamical systems. Notably, our framework enables the analysis to be made efficiently and conveniently over a parameter range. As such, it becomes an essential tool for stability analysis of dynamical systems that goes beyond local stability analysis offered by alternative frameworks. We demonstrate the effectiveness of our approach on a variety of models, including climate, power grids, ecosystems, and more. Our framework is available as simple-to-use open-source code as part of the DynamicalSystems.jl library.