中科院数学与系统科学研究院

数学研究所

学术报告

动力系统研讨班

报告人： 王式柔 副教授（吉林大学）   
题  目：Synchronization in Markov random networks
时  间：2022.05.18（星期三）上午9:00-10:00
地  点：腾讯会议：532-920-864 密码：0518
摘  要：Many complex biological and physical networks are naturally subject to both random influences, i.e., extrinsic randomness, from their surrounding environment, and uncertainties, i.e., intrinsic noise, from their individuals. Among many interesting network dynamics, of particular importance is the synchronization property which is closely related to the network reliability especially in cellular bio-networks. It has been speculated that whereas extrinsic randomness may cause noise-induced synchronization, intrinsic noises can drive synchronized individuals apart. This talk presents an appropriate framework of (discrete state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the study of such synchronization and desynchronization scenario. By studying the asymptotics of the Markov perturbed stationary distributions, probabilistic characterizations of the alternating pattern between synchronization and desynchronization behaviours is given.  It is shown that if a random network without intrinsic noise is synchronized, then after intrinsic noise perturbation, high-probability synchronization and low-probability desynchronization can occur intermittently and alternatively in time, and moreover, both the probability of (de)synchronization and the proportion of time spent in (de)synchrony can be explicitly estimated. Further problems related to this topic will also be discussed.

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报告人： 黄博（北京航空航天大学）   
题  目：On the Number of Limit Cycles from Zero-Hopf Bifurcation for Certain Differential Systems
时  间：2022.05.18（星期三）上午10:30-11:30
地  点：数学院南楼N913 腾讯会议：532-920-864 密码：0518
摘  要：In this talk, I will introduce some new results on the number of limit cycles from zero-Hopf bifurcation for certain differential systems. The system in question is assumed to be of dimension $n$, have a zero-Hopf equilibrium at the origin, and consist only of homogeneous terms of order $m$. Using the averaging method, we obtain some upper bounds for the number of limit cycles for generic $n\geq3$ and $m\geq2$. The exact numbers of limit cycles or tight bounds on the numbers are determined by computing the mixed volumes of some polynomial systems obtained from the averaged functions. Based on symbolic computation, a general and algorithmic approach is proposed to derive sufficient conditions for a given differential system to have a prescribed number of limit cycles. The effectiveness of the proposed approach is illustrated by a family of third-order differential equations and by a four-dimensional hyperchaotic differential system.    This is based on my joint work with Prof. Dongming Wang.