Framework for global stability analysis of dynamical systems

What is it about?

Dynamical systems are used to model power grids, the brain, climate, and many other natural systems. Most dynamical systems display multistabilitiy, which means that there could be several coexisting stable states, also known as "attractors". For example, the glacial "ice-ages" and the current "warm Earth" can be viewed as two attractors of the Earth climate system. Our work provides a computational framework to analyze such multistable dynamical systems, in terms of the global (or basin) stability of the individual attractors in the system. The framework is accompanied by an easy to use high-performance code base to perform such analysis within only a couple lines of code.

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Photo by Daniele Levis Pelusi on Unsplash

Photo by Daniele Levis Pelusi on Unsplash

Why is it important?

Global (or basin) stability is very important for the concept of tipping points: transitions of a dynamical system from one attractor to another. There, it is important to quantify how the system may respond to finite perturbations. Insofar, the only accessible way to characterize stability of dynamical systems was via the local stability, which only describes the response of the system to infinitesimal perturbations. However, for studying tipping points, and other related phenomena, one needs to know how the system responds to finite perturbations. This is the kind of information that global stability offers. Hence, our framework complements this established standard and allows expanding stability analysis into a realm that deals with finite perturbations. We believe our work will accelerate research on multistability and tipping points.

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