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

@article{Kuehn2011AMF,
  title={A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications},
  author={Christian Kuehn},
  journal={Journal of Nonlinear Science},
  year={2011},
  volume={23},
  pages={457-510}
}
  • C. Kuehn
  • Published 14 January 2011
  • Mathematics
  • Journal of Nonlinear Science
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… 

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References

SHOWING 1-10 OF 163 REFERENCES

A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics

Early warning signs for saddle-escape transitions in complex networks

It is shown that in high-dimensional systems, containing many variables, the authors frequently encounter an additional non-bifurcative saddle-type mechanism leading to critical transitions, which establishes a connection from critical transitions to networks and an early warning sign for a new type of critical transition.

Noise-Induced Phenomena in Slow-Fast Dynamical Systems: A Sample-Paths Approach

Stochastic differential equations play an increasingly important role in modeling the dynamics of a large variety of systems in the natural sciences, and in technological applications. This book is

Impact of noise on bistable ecological systems

Metastability in simple climate models: Pathwise analysis of slowly driven Langevin equations

We consider simple stochastic climate models, described by slowly time-dependent Langevin equations. We show that when the noise intensity is not too large, these systems can spend substantial

Climate tipping as a noisy bifurcation: a predictive technique

It is often known, from modelling studies, that a certain mode of climate tipping (of the oceanic thermohaline circulation, for example) is governed by an underlying fold bifurcation. For such a case

Early warning signals of extinction in deteriorating environments

It is argued that the causes of a population’s decline are central to the predictability of its extinction, and populations crossing a transcritical bifurcation, experimentally induced by the controlled decline in environmental conditions, show statistical signatures of CSD after the onset of environmental deterioration and before the critical transition.

Generalized models as a universal approach to the analysis of nonlinear dynamical systems.

  • Thilo GrossU. Feudel
  • Mathematics
    Physical review. E, Statistical, nonlinear, and soft matter physics
  • 2006
The proposed approach allows us to study the dynamical properties of generalized models efficiently in the framework of local bifurcation theory and yields a close connection between modelling and nonlinear dynamics.

Spatial correlation as leading indicator of catastrophic shifts

Generic early-warning signals such as increased autocorrelation and variance have been demonstrated in time-series of systems with alternative stable states approaching a critical transition.

Elements of Mathematical Ecology

  • M. Kot
  • Mathematics, Environmental Science
  • 2001
Preface Part I. Unstructured Population Models Section A. Single Species Models: 1. Exponential, logistic and Gompertz growth 2. Harvest models - bifurcations and breakpoints 3. Stochastic birth and
...