A Probabilistic Distance-Based Stability Quantifier for Complex Dynamical Systems

Abstract

An attractor of a dynamical system may represent the system's 'desirable' state. Perturbations to the system may push the system out of the basin of attraction of the desirable attractor and into undesirable states. Hence, it is important to quantify the stability of such systems against reasonably large perturbations. In this paper, we introduce a distance-based measure of stability, called 'basin stability bound', to characterise the stability of dynamical systems against finite perturbations. This stability measure depends on both the size and the shape of the basin of attraction of the desirable attractor. A probabilistic sampling-based approach is used to estimate basin stability bound and quantify the associated estimation error. This approach allows for the easy estimation of basin stability bound regardless of the structure of the basin of attraction and is readily applicable to high-dimensional systems. We demonstrate the merit of the proposed stability measure using an ecological model of the Amazon rainforest, a ship capsize model, and a power grid model.