珠海数学杰出学者讲座第36讲-HOPF BIFURCATION OF LIMIT CYCLES SURROUNDING A NODE

 讲座题目：HOPF BIFURCATION OF LIMIT CYCLES SURROUNDING A NODE
讲座时间 Datetime: 2024年10月11日，周五，9:00-12:00
地点 Venue: 海琴二号457
报告人Speaker: Jaume Llibre教授
单位 Affiliation: 巴塞罗那自治大学
主持人 Host：赵育林 教授
报告摘要 Abstract:
We consider the planar polynomial differential systems in R2
x˙ = µx + P (x, y, µ),
y˙ = µy + Q(x, y, µ),
where the polynomials P and Q have neither constant terms nor linear terms, satisfying that when the parameter |µ| ̸= 0 and small, the origin is a node, and for µ = 0 the origin is either a node or a focus. The main theorem characterizes the Hopf bifurcation from the equilibrium point at the origin of this system. We illustrate this result with several examples. As a consequence, we can give a lower bound for the number of small limit cycles surrounding a node depending only on the degree of the polynomial differential system. In summary, we extend the Hopf bifurcation that usually studies the bifurcation of limit cycles from a focus to the bifurcation of limit cycles from a node.
In this talk the node at the origin is an star node, but the results that I will present for the star node also work for the standard node ˙ x = µx, y˙ = λy with µλ > 0, or for the non-diagonalizable node ˙ x = µx + y, y˙ = µy.
报告人简介：
Jaume Llibre教授是巴塞罗那自治大学数学系教授，西班牙皇家科学学院院士，是常微分方程与动力系统领域的国际知名专家，领导着一个十分活跃的研究团队，目前担任国际SCI杂志《Qualitative Theory of Dynamical Systems》的主编，同时也是多个国际SCI杂志的编委，在该领域具有世界领先的学术研究水平和学术影响。