Mathematics – Dynamical Systems
Scientific paper
2011-01-14
Mathematics
Dynamical Systems
Improved notation and some explanations in comparison to previous version
Scientific paper
Critical transitions occur in a wide variety of applications including mathematical biology, climate change, human physiology and economics. Therefore it is highly desirable to find early-warning signs. In this article we show that it is straightforward to classify critical transitions by using bifurcation theory and normal forms. Based on this classification, we analyze stochastic fluctuations and calculate scaling laws of the variance of stochastic sample paths near critical transitions for fast subsystem bifurcations up to codimension two. The theory is applied 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. In each model we find new results that are relevant from a theoretical as well as an applied perspective.
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