title: Towards a critical transition theory under different temporal scales and noise strengths.
abstract: The mechanism of critical phenomena or critical transitions has been recently studied from various aspects, in particular considering slow parameter change and small noise. In this article, we systematically classify critical transitions into three types based on temporal scales and noise strengths of dynamical systems. Specifically, the classification is made by comparing three important time scales τ(λ), τ(tran), and τ(ergo), where τ(λ) is the time scale of parameter change (e.g., the change of environment), τ(tran) is the time scale when a particle or state transits from a metastable state into another, and τ(ergo) is the time scale when the system becomes ergodic. According to the time scales, we classify the critical transition behaviors as three types, i.e., state transition, basin transition, and distribution transition. Moreover, for each type of transition, there are two cases, i.e., single-trajectory transition and multitrajectory ensemble transition, which correspond to the transition of individual behavior and population behavior, respectively. We also define the critical point for each type of critical transition, derive several properties, and further propose the indicators for predicting critical transitions with numerical simulations. In addition, we show that the noise-to-signal ratio is effective to make the classification of critical transitions for real systems.
authors.affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China.
authors.forename: Jifan
authors.initials: J
authors.lastname: Shi
authors.affiliation: LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, China.
authors.forename: Tiejun
authors.initials: T
authors.lastname: Li
authors.affiliation: Key Laboratory of Systems Biology, Innovation Center for Cell Signaling Network, Institute of Biochemistry and Cell Biology, Shanghai Institutes for Biological Sciences, and Chinese Academy of Sciences, Shanghai 200031, China; Institute of Industrial Science, University of Tokyo, Tokyo 153-8505, Japan.
authors.forename: Luonan
authors.initials: L
authors.lastname: Chen