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How basin stability complements the linear-stability paradigm

Nature Physics volume 9, pages 89–92 (2013)Cite this article

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Abstract

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 topologies8,9,10 of neural networks and power grids, which have eluded theoretical description based solely on linear stability11,12,13. We anticipate that basin stability will provide a powerful tool for complex systems studies, including the assessment of multistable climatic tipping elements14.

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Figure 1: Thought experiment: marble on a marble track.
Figure 2: State diagram of a bistable stylized forest–savanna model.
Figure 3: Synchronizability and basin stability in Watts–Strogatz networks of chaotic oscillators.
Figure 4: Topological comparison of ensemble results with real-world networks.

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Acknowledgements

The authors acknowledge financial support from IRTG 1740 (Deutsche Forschungsgemeinschaft), the SUMO-project (European Union), the ECONS-project (Leibniz Association) and Konrad-Adenauer-Stiftung. They thank N. Fujiwara and A. Rammig for inspiring discussions.

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Authors and Affiliations

  1. Research Domain on Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, PO Box 60 12 03, 14412 Potsdam, Germany

    Peter J. Menck, Jobst Heitzig, Norbert Marwan & Jürgen Kurths

  2. Department of Physics, Humboldt University of Berlin, Newtonstraße 15, 12489 Berlin, Germany

    Peter J. Menck & Jürgen Kurths

  3. Institute for Complex Systems and Mathematical Biology, University of Aberdeen, Aberdeen AB24 3UE, UK

    Jürgen Kurths

Authors
  1. Peter J. Menck
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  2. Jobst Heitzig
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  3. Norbert Marwan
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  4. Jürgen Kurths
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Contributions

P.J.M. conceived the study, performed the numerics, and prepared the manuscript. All authors discussed the results, drew conclusions and edited the manuscript. J.K. supervised the study.

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Correspondence to Peter J. Menck.

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The authors declare no competing financial interests.

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Menck, P., Heitzig, J., Marwan, N. et al. How basin stability complements the linear-stability paradigm. Nature Phys 9, 89–92 (2013). https://doi.org/10.1038/nphys2516

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