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

@article{Kuehn2013AMF,
  title={A Mathematical Framework for Critical Transitions: Normal Forms, Variance and Applications},
  author={Christian Kuehn},
  journal={Journal of Nonlinear Science},
  year={2013},
  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… 

Early-warning signs for pattern-formation in stochastic partial differential equations

Warning signs for non-Markovian bifurcations: colour blindness and scaling laws

This work significantly advance the general theory of warning signs for nonlinear stochastic dynamics by considering different types of underlying noise, including colored noise and α-regular Volterra processes.

Predictability of critical transitions.

This contribution investigates the predictability of critical transitions in conceptual models using the quadratic integrate-and-fire model and the van der Pol model under the influence of external noise and focuses especially on the statistical analysis of the success of predictions and the overall predictability.

Uncertainty Quantification of Bifurcations in Random Ordinary Differential Equations

A methodology to determine the probability of the occurrence of different types of bifurcations based on the probability distribution of the input parameters and captures the major qualitative behavior of the RODEs.

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.

Nonlocal generalized models of predator-prey systems

This paper analyzes predator-prey dynamical systems and extends the method of generalized models to periodic solutions to adapt the equilibrium generalized modeling approach and compute the unique Floquet multiplier of the periodic solution which depends upon so-called generalized elasticity and scale functions.

Linear stability theory as an early warning sign for transitions in high dimensional complex systems

We analyse in detail a new approach to the monitoring and forecasting of the onset of transitions in high dimensional complex systems by application to the Tangled Nature model of evolutionary

Robustness of early warning signals for catastrophic and non-catastrophic transitions

It is found that EWS generally work well to signal an impending saddle-node bifurcation, regardless of the autocorrelation or intensity of the noise, but different methods should be used to predict other types of regime shifts.

Beyond bifurcation: using complex models to understand and predict abrupt climate change

This paper focuses on several climate system components and ecosystems that have been proposed as candidates for tipping points, with an emphasis on ice sheets, the Atlantic Ocean circulation, vegetation in North Africa and Arctic sea ice and the potential of data mining approaches to tackle this problem.
...

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.
...