How basin stability complements the linear-stability paradigm

The human brain1,2, power grids3, arrays of coupled lasers4 and the Amazon rainforest5,6 are all characterized by multistability7. The likelihood that these systems will remain in the most desirable of their many stable states depends on their stability against significant perturbations, particularly in a state space populated by undesirable states. Here we claim that the traditional linearization-based approach to stability is too local to adequately assess how stable a state is. Instead, we quantify it in terms of basin stability, a new measure related to the volume of the basin of attraction. Basin stability is non-local, nonlinear and easily applicable, even to high-dimensional systems. It provides a long-sought-after explanation for the surprisingly regular topologies8–10 of neural networks and power grids, which have eluded theoretical description based solely on linear stability11–13. We anticipate that basin stability will provide a powerful tool for complex systems studies, including the assessment of multistable climatic tipping elements14. Complex systems science relies heavily on linear stability analysis, in which state of a dynamic system (more correctly, its dynamic regime) is assessed basically by inspecting the dominant curvature of the potential energy function in the state’s surroundings (as expressed by Lyapunov exponents). The absolute value of the curvature measures the speed of convergence or divergence after a small perturbation, and its sign qualifies the state as stable or unstable. Such linearization-based considerations are inherently local; therefore, they are not sufficient to quantify how stable a state is against non-small perturbations. Quantification of stability in this sense requires a global concept: the basin of attraction B of a state is the set of initial points in state space from which the system converges to this state. Complete knowledge of the basin would allow us to fully assess the state’s stability: one could classify perturbations into the permissible and the impermissible. See Fig. 1. However, basins are intricate entities15 and especially hard to explore in high dimensions. Here we therefore focus on a single but fundamental property: the basin’s volume. The authors of ref. 16 interpret the volume of a state’s basin of attraction as a measure of the likelihood of arrival at this state, that is, as a measure of the state’s relevance. Almost equivalently, we understand the volume of the basin as an expression of the likelihood of return to the state after any random—possibly non-small—perturbation. This yields a second interpretation: the basin’s volume quantifies how stable a state is. To the best of our knowledge, this interpretation has not yet been employed in complex systems science.We refer to the quantification of stability based on the basin’s volume as basin stability SB. For climatic tipping elements14 it would be particularly useful to know how stable the desirable (that is, present) state is against