A mathematical framework for critical transitions: Bifurcations, fastslow systems and stochastic dynamics
Abstract
Bifurcations can cause dynamical systems with slowly varying parameters to transition to faraway 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 earlywarning 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 fastslow 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 fastslow systems. The interplay between sample path techniques, partial differential equations and random dynamical systems is highlighted. Each viewpoint provides potential earlywarning signs for critical transitions. Since increasing variance has been suggested as an earlywarning 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 earlywarning signs for generic models. We end with suggestions for future directions of the theory.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 June 2011
 DOI:
 10.1016/j.physd.2011.02.012
 arXiv:
 arXiv:1101.2899
 Bibcode:
 2011PhyD..240.1020K
 Keywords:

 Mathematics  Dynamical Systems;
 Mathematics  Classical Analysis and ODEs;
 Nonlinear Sciences  Chaotic Dynamics;
 Nonlinear Sciences  Pattern Formation and Solitons
 EPrint:
 38 pages