Journal of Nonlinear Science

, Volume 23, Issue 3, pp 457–510 | Cite as

A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications

  • Christian Kuehn


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. We show that it is possible to classify critical transitions by using bifurcation theory and normal forms in the singular limit. Based on this elementary 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. For the Stommel–Cessi model we compare different detrending techniques to calculate early-warning signs. In the epidemics model we show that link densities could be better variables for prediction than population densities. The activator–inhibitor switch demonstrates effects in three time-scale systems and points out that excitable cells and molecular units have information for subthreshold prediction. In the predator–prey model explosive population growth near a codimension-two bifurcation is investigated and we show that early-warnings from normal forms can be misleading in this context. In the biomechanical model we demonstrate that early-warning signs for buckling depend crucially on the control strategy near the instability which illustrates the effect of multiplicative noise.


Critical transition Tipping point Fast-slow system Invariant manifold Stochastic differential equation Multiple time scales Moment estimates Asymptotic analysis Laplace integral Thermohaline circulation Activator–inhibitor system Adaptive networks SIS-epidemics Bazykin predator–prey model Euler buckling 

Mathematics Subject Classification

34F05 34E15 60H10 



I would like to thank Martin Zumsande for suggesting the model from systems biology in Sect. 7.3 and Thilo Gross for insightful discussions regarding network dynamics. I also would like to thank two anonymous referees and the editor for many helpful comments that helped to improve the manuscript. Part of this work was supported by the European Commission (EC/REA) via a Marie-Curie International Re-integration Grant.


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Max Planck InstitutePhysics of Complex SystemsDresdenGermany
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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