威尼斯人

报告题目: On Asymptotic Measure of Quasi-Potential Systems with Applications to Stochastic Bifurcations

报告人：蒋继发(上海师范大学)

报告时间：2022年12月03日09:00-09:50

报告地点：腾讯会议：693-383-455

会议密码请邮件咨询yangdw@vnsclub.net

报告摘要：The asymptotic measure of stationary measures for a gradient system under small additive noise perturbation is concentrated on the global minima set of the potential function.  Hwang (1980) and Huang et al.(2016) determined the weights of the asymptotic measure supported on equilibria if the global minima set consists of only finite points at which the first nonzero homogeneous polynomials of Taylor expansions are even order. This talk will focus on the asymptotic measure problem of a quasi-potential system which is the sum of a gradient system and a system orthogonal to the gradient system.  It is proved that stationary measures of the gradient system and the quasi-potential system under the same additive noise perturbation admit the same density if the divergence of the orthogonal system is zero. We also provide orthogonal group invariance criterion of the  density of stationary measure, which helps us to determine the supporting components' weights. Combining these with the global dynamics of quasi-potential system, we give exact asymptotic measure and its support, including stable equilibria, stable periodic orbits, saddles, and chaotic motions et al., of  a large number of quasi-potential systems, therefore, gradient systems with continuum of the global minima set under the same additive noise perturbation.  Besides, we study stochastic bifurcations of two or three dimensional quasi-potential systems using our criteria.