【海外学术大师】关于举行易英飞教授(阿尔伯塔大学)短期讲学的通知

报告主题: Topic I.  Synchronization in Markov random networks

     Topic II. Transient Behaviors in Dynamical Systems  

     Topic III: Response Solution and Ergodicity

报  告  人: 易英飞 教授 

报告时间：2024年4月8-10日（具体见下表）

报告地点：清清文理楼3A02  

 邀  请  人: 杨启贵 教授

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数学学院

2024年4月2日

报告人简介：易英飞，教授，国家级高层次人才，吉林大学柔性引进人才特聘教授，阿尔伯塔大学Killiam讲座教授，佐治亚理工学院兼职教授，Journal of Dynamics and Differential Equations杂志主编、Journal of Differential Equations等杂志的编委，研究方向为动力系统、复杂系统等动力学基础理论。在国际期刊发表文章80 余篇，引用 2000余次。

报告摘要：

Topic I.  Synchronization in Markov random networks
Abstract: 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 series of lectures presents an appropriate framework of (discrete-state and discrete time) Markov random networks to incorporate both extrinsic randomness and intrinsic noise into the rigorous 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 behaviors is given.  More precisely, it is shown that if a random network without intrinsic noise perturbation 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. 

Topic II. Transient Behaviors in Dynamical Systems
Abstract:  Transient dynamics are those display and being almost invariant over finite but very long time interval. They often arise in multi-scale systems and represent true physical stationary dynamics. Well-known examples in deterministic systems include transient oscillations and transient chaos. For stochastic systems, an important example is the quasi-stationary distributions (QSDs) which are almost invariant to a diffusion process over exponentially long time. Representing important transient stochastic dynamics, QSDs arise frequently in applications especially in chemical reactions and population systems admitting extinction states. This series of lectures will present some rigorous results on transient oscillations and QSDs. For the former, a pattern of multiple transients will be shown, and for the later, the existence, uniqueness, concentration, and convergence of QSDs will be shown along with their connections to the spectrum of the Fokker-Planck 

Topic III: Response Solution and Ergodicity

Abstract.  In a non-autonomous dynamical system, response solutions are those corresponding to the structure of the time dependence. For instance, in a quasi-periodically forced differential equation, response solutions are quasi-periodic ones whose frequency vector coincides with that of the forcing function. These solutions are known to play a fundamental role in the harmonic and synchronizing behaviors of periodically or quasi-periodically forced oscillators. For stochastic systems with periodic  coefficients, response solutions are time-periodic distributions whose period is the same as that of the coefficients. These solutions are known to closely related to ergodic behaviors of the system. This series of lectures will present some recent results on the existence of responsive solutions in degenerate, quasi-periodically forced nonlinear oscillators with small or free damping. The cases with singular or noise perturbations will also be discussed. It will also give some discussions about ergodicity of stochastic systems, especially for mesoscopic systems with less regular coefficients, for both autonomous and time-periodic cases.