报告题目：Homoclinic Bifurcations with Applications to Biological Systems

报告人：Shigui Ruan，University of Miami

邀请人：刘丹

报告时间：2023年11月14日（周二）16:00-17：00

报告地点：腾讯会议624-729-084

报告人简介：Shigui Ruan，美国迈阿密大学数学系教授和Cooper Fellow。主要研究领域是动力系统和微分方程及其在生物和医学中的应用，在PNAS、Memoirs Amer Math Soc、J Math Pures Appl、Math Ann等学术期刊上发表论文两百多篇，受到国际同行的关注与大量引用，2013年获得海外及港澳学者合作研究基金资助，2015、2016连续两年入选汤森路透全球高被引科学家。对一些传染病（如非典、乙型肝炎、血吸虫感染、狂犬病、疟疾、院内感染等）的数学建模、数据模拟和理论分析作出了一系列开创性工作，不仅填补了该领域的科研空白，而且对理解和控制这些传染病在中国的流行和传播有着非常重要的现实意义。担任生物数学领域重要期刊《Mathematical Biosciences》、《DCDS-B》、《BMC Infectious Diseases》、《Bulletin of Mathematical Biology》、《Journal of Biological Systems》与《International Journal of Biomathematics》等编委，是《Mathematical Biosciences and Engineering》的主编（数学）。作为项目负责人多次获得美国国家卫生研究院（NIH）、美国国家科学基金（NSF）、国家自然科学基金会资助。

报告摘要：For n-dimensional differential equations (n≥2), a homoclinic orbit associated to a hyperbolic equilibrium point is an orbit that has the point as its α-limit set as well as its ω-limit set. Since a homoclinic orbit is structurally unstable, bifurcation occurs when the homoclinic loop is broken. There are two approaches to study such bifurcations: Melnikov function and Shilnikov approach. In this talk, we introduce Shilnikov method in studying homoclinic bifurcations. For planar systems, a homoclinic orbit associated to a saddle will induce a limit cycle under certain condition. For 3-dimensional systems, there are two types of hyperbolic equilibrium points: saddle and saddle-focus. Under various conditions, a homoclinic orbit associated to a saddle will bifurcate to a (stable or saddle) cycle, whereas a homoclinic orbit associated to a saddle-focus will bifurcate to either a stable cycle or infinitely many saddle cycles (that is, chaos) via Smale horseshoe map. Applications to homoclinic bifurcations and chaos in various biological models, including FitzHugh–Nagumo neuron model, three-species food chain models, green ocean plankton models, epileptic models, and chemostat models, will be presented. Generalizations to infinite dimensional systems will be briefly discussed.