Formal Measures of Dynamical Properties: Tipping Points, Robustness, and Sustainability

  • Aaron BramsonEmail author
Part of the Understanding Complex Systems book series (UCS)


Complex systems are characterized by processes that exhibit feedback, nonlinearity, heterogeneity, and path dependencies, and accurately modeling such systems is becoming increasing important. To help realize the potential of complex systems modeling we need new methods that are capable of capturing the dynamical properties of such processes across disciplines and modeling frameworks. This chapter presents a portion of the methodology development that includes formal and domain-agnostic definitions of phenomena related to tipping points, criticality, robustness, and sustainability. For each included concept I provide a probabilistic definition based on a Markov model generated from time-series data in a specific way. These rigorous mathematical definitions clearly distinguish multiple distinct dynamical properties related to each concept, and they also function as measures of these properties. Though only a small portion of the methodology’s capabilities, theorems, and applications can be included in this treatment, it does include all the foundational material necessary to apply the methodology.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Laboratory for Symbolic Cognitive DevelopmentCenter for Biosystems Dynamics ResearchWako CityJapan
  2. 2.Department of General EconomicsGhent UniversityGhentBelgium

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