Formal Measures of Dynamical Properties: Tipping Points

To help realize the potential of complex systems models we need new measures appropriate for capturing processes that exhibit feedback, nonlinearity, heterogeneity, and emergence. As part of a larger research project encompassing several categories of dynamical properties this paper provides formal and general definitions of tipping point-related phenomena. For each tipping concept this paper provides a probabilistic definition derived from a Markov model representation. We start with the basic features of Markov models and definitions of the foundational concepts of system dynamics. Then several tipping point-related concepts are described, defined, measured, and illustrated with a simplified graphical example. The paper finishes with several branches of future work involving new measures for complex systems and the fusion of research domains.