Identification of unstable fixed points for randomly perturbed dynamical systems with multistability

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:446

Article

Chen, Xian¹  Jia, Chen^2,3 

[1] Peking Univ, Sch Math Sci, Beijing 100871, Peoples R China

[2] Beijing Computat Sci Res Ctr, Beijing 100094, Peoples R China

[3] Univ Texas Dallas, Dept Math Sci, Dallas, TX 75080 USA

关键词: Bistability;    Diffusion process;    Downhill timescale;    Maximum inequality;   

DOI  :  10.1016/j.jmaa.2016.07.060

来源: Elsevier

【 摘 要 】

Multistability, especially bistability, is one of the most important nonlinear phenomena in deterministic and stochastic dynamics. The identification of unstable fixed points for randomly perturbed dynamical systems with multistability has drawn increasing attention in recent years. In this paper, we provide a rigorous mathematical theory of the previously proposed data-driven method to identify the unstable fixed points of multistable systems. Specifically, we define a family of statistics which can be estimated by practical time-series data and prove that the local maxima of this family of statistics will converge to the unstable fixed points asymptotically. During the proof of the above result, we obtain two mathematical by-products which are interesting in their own right. We prove that the downhill timescale for randomly perturbed dynamical systems is log(1/epsilon), different from the uphill timescale of e(v/epsilon) for some V > 0 predicted by the Freidlin-Wentzell theory. Moreover, we also obtain an L-P maximum inequality for randomly perturbed dynamical systems and a class of diffusion processes. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free