A mathematical framework for critical transitions: Bifurcations, fast–slow systems and stochastic dynamics
By Christian Kuehn. 2011
415 citations recorded by the source corpus.
Abstract Bifurcations can cause dynamical systems with slowly varying parameters to transition to far-away attractors. The terms “critical transition” or “tipping point” have been used to describe this situation. Critical transitions have been observed in an astonishingly diverse set of applications from ecosystems and climate change to medicine and finance. The main goal of this paper is to give an overview which standard mathematical theories can be applied to critical transitions. We shall focus on early-warning signs that have been suggested to predict critical transitions and point out what mathematical theory can provide in this context. Starting from classical bifurcation theory and incorporating multiple time scale dynamics one can give a detailed analysis of local bifurcations that induce critical transitions. We suggest that the mathematical theory of fast–slow systems provides a natural definition of critical transitions. Since noise often plays a crucial role near critical transitions the next step is to consider stochastic fast–slow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential early-warning signs for critical transitions. Since increasing variance has been suggested as an early-warning sign we examine it in the context of normal forms analytically, numerically and geometrically; we also consider autocorrelation numerically. Hence we demonstrate the applicability of early-warning signs for generic models. We end with suggestions for future directions of the theory.
Subfields: Atmospheric Science, Global and Planetary Change, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/c722438db3bff6931b2788f44ff05bb3a7fb7ff4
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Atmospheric Science
- Vasilis Dakos
- Marten Scheffer
- Sonia Kefi
- Stephen R. Carpenter
- M. Rietkerk
- E. V. van Nes
- Christian Kuehn
In Global and Planetary Change
- Vasilis Dakos
- Stephen R. Carpenter
- E. V. van Nes
- Marten Scheffer
- Sonia Kefi
- M. Rietkerk
- Carl Folke
- Christian Kuehn
In Statistical and Nonlinear Physics
- Jan Pukropski
- Marten Scheffer
- R. von Wrede
- Sonia Kefi
- Klaus Lehnertz
- T. Rings
- Vasilis Dakos
- E. V. van Nes
Citations in the corpus
- Early warning signals also precede non-catastrophic transitions (2013, background): “However, the mathematical properties of a system going through these types of critical transitions (Kuehn 2011) is largely independent of the specific model formulation.”
- Resilience indicators: prospects and limitations for early warnings of regime shifts (2015, background): “Critical transitions are defined as abrupt qualitative changes in the state of an ecosystem that occur close to bifurcation points [21].”
- Tipping points in open systems: bifurcation, noise-induced and rate-dependent examples in the climate system (2011, neutral): “Keywords: rate-dependent tipping point; bifurcation; climate system”
- The time-evolving epileptic brain network: concepts, definitions, accomplishments, perspectives (2024, background): “…to the different types of intermittency (Perez Velazquez et al., 1999; Rizzi et al., 2016; Pisarchik et al., 2018), switching in fast-slow systems (Kuehn, 2011) and in heteroclinic networks (Kirst and Timme, 2008; Aguiar et al., 2011; Bick and Field, 2017; Morrison and Young, 2022;…”