title: Stochastic basins of attraction for metastable states.
abstract: Basin of attraction of a stable equilibrium point is an effective concept for stability analysis in deterministic systems; however, it does not contain information on the external perturbations that may affect it. Here we introduce the concept of stochastic basin of attraction (SBA) by incorporating a suitable probabilistic notion of basin. We define criteria for the size of the SBA based on the escape probability, which is one of the deterministic quantities that carry dynamical information and can be used to quantify dynamical behavior of the corresponding stochastic basin of attraction. SBA is an efficient tool to describe the metastable phenomena complementing the known exit time, escape probability, or relaxation time. Moreover, the geometric structure of SBA gives additional insight into the system's dynamical behavior, which is important for theoretical and practical reasons. This concept can be used not only in models with small noise intensity but also with noise whose amplitude is proportional or in general is a function of an order parameter. As an application of our main results, we analyze a three potential well system perturbed by two types of noise: Brownian motion and non-Gaussian α-stable Lévy motion. Our main conclusions are that the thermal fluctuations stabilize the metastable system with an asymmetric three-well potential but have the opposite effect for a symmetric one. For Lévy noise with larger jumps and lower jump frequencies ( α=0.5) metastability is enhanced for both symmetric and asymmetric potentials.
authors.affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China and Department of Science and Technology, University of Cape Verde, Praia 7600, Cape Verde.
authors.forename: Larissa
authors.initials: L
authors.lastname: Serdukova
authors.affiliation: School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
authors.forename: Yayun
authors.initials: Y
authors.lastname: Zheng
authors.affiliation: Department of Applied Mathematics, Illinois Institute of Technology, Chicago, Illinois 60616, USA and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China.
authors.forename: Jinqiao
authors.initials: J
authors.lastname: Duan
authors.affiliation: Research Domain on Transdisciplinary Concepts and Methods, Potsdam Institute for Climate Impact Research, P.O. Box 601203, 14412 Potsdam, Germany and Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan 430074, China.
authors.forename: Jürgen
authors.initials: J
authors.lastname: Kurths