BioND — Dynamics of Biological Networks

A mathematical framework for critical transitions: Normal forms, variance, and applications

Christian Kuehn
Journal of Nonlinear Science 23(3), 457-510, 2013.
arXiv:1101.2908

Abstract

Bifurcation Diagram

Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. A dynamical system that was in a stable state suddenly changes to a distant attractor. Therefore it is highly desirable to find early-warning signs for these critical transitions. Although several different approaches have been proposed for specific models, a detailed mathematical theory has not been developed yet. In this paper we provide another building block of this theory beyond the first results developed in [C. Kuehn. "A mathematical framework for critical transitions: bifurcations, fast-slow systems and stochastic dynamics", submitted, 2010]. We start from classical bifurcation theory and normal forms to classify critical transitions. Using this approach, we proceed to stochastic fluctuations and provided detailed scaling laws of the variance of stochastic sample paths near critical transitions. To show that our theory naturally extends bifurcation theory we apply it to several models: the Stommel-Cessi box model for the thermohaline circulation from geoscience, an epidemic-spreading model on an adaptive network, an activator-inhibitor switch from systems biology, a predator-prey system from ecology and to the Euler buckling problem from classical mechanics. We also hope that our combination of theory and applications for critical transitions opens up new research directions in different disciplines of nonlinear science.

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