报告题目1：Bifurcations of $Z_q$-equivariant vector fields and possible configurations of limit cycles

报告人：李继彬教授，华侨大学

报告时间：2024年11月5日（周二）14:00-15:00

报告地点：22-214

报告摘要：Like the first part of Hilbert's 16th problem where the distribution of ovals is to be considered, the distribution problem of limit cycles can also be very interesting. Coleman [1983] in his survey “Hilbert 16th problem: How Many Cycles?” stated that “For n>2 the maximal number of eyes is not known, nor is it known just which complex patterns of eyes within eyes, or eyes enclosing more than a single critical point, can exist. Here so-called “eye” means the limit cycle. In order to obtain more limit cycles and various configuration patterns of their relative dispositions, we indicated that an efficient method is to perturb the symmetric Hamiltonian systems having maximal number of centers, i.e. to study the weakened Hilbert's 16th problem for the symmetric planar polynomial Hamiltonian systems, since bifurcation and symmetry are closely connected and symmetric systems play pivotal roles as a bifurcation point in all planar Hamiltonian system class. To investigate perturbed Hamiltonian systems, we should first know the global behavior of unperturbed polynomial systems, namely, determine the global property for the families of real planar algebraic curves defined by the Hamiltonian functions. Then by using proper perturbation techniques, we shall obtain the global information of bifurcations for the perturbed nonintegrable systems. In this talk, we introduce the method of detection functions posed by Li et al. [1985], to study the $Z_q$-equivariant perturbed polynomial systems and the method of control parameters. We also present recent developments in this direction. 

报告人简介：李继彬，华侨大学教授，博士生导师。曾任四届国家自然科学基金委数学学科评审专家组成员，云南省科学技术委员会常务委员，三届云南省数学会理事长，云南省应用数学研究所副所长，昆明理工大学理学院院长等。现为《应用数学与力学》等全国和国际性刊物的编委；美国《数学评论》与德国《数学文摘》评论员。主持承担国家自然科学基金重点项目和面上项目等10余项，发表论文250余篇，出版中英文专著10部，主编教材2部、出版科普书2本。三十余年培养硕士和博士研究生70余人。科研成果曾分别获云南省和浙江省科学技术一等奖。

报告题目2：Poincaré compactification for $n$-dimensional piecewise polynomial vector fields: Theory and Applications

报告人：李时敏教授，杭州师范大学

报告时间：2024年11月5日（周二）15:00-16:00

报告地点：22-214

报告摘要：In this talk we will extend the Poincaré compactification to $n$-dimensional piecewise polynomial vector field, Thus we can investigate the dynamics of the infinity of n-dimensional piecewise polynomial vector fields. The main differences of Poincare compactification between polynomial vector fields and piecewise polynomial vector fields are concluded. As an application, we study the global phase portraits for a class of 3-dimensional piecewise linear differential systems.

报告人简介：李时敏，杭州师范大学教授，理学博士，硕士生导师。研究方向为微分方程理论及其应用，主要研究希尔伯特第十六问题及其相关问题。在中国科学.数学，J. Differential Equations，J. Nonlinear Science等国内外学术期刊上发表论文30余篇，研究成果获广东省自然科学奖二等奖（排名第四）。先后主持国家自然科学基金面上项目和青年项目各1项，广东省自然科学基金2项，广州市科技创新研究专项项目1项。

报告题目3：Bifurcation of Limit cycles from a heteroclinic loop connecting a tangent non-Morse point and a hyperbolic saddle

报告人：孙宪波教授，杭州师范大学

报告时间：2024年11月5日（周二）16:00-17:00

报告地点：22-214

报告摘要：We study the limit cycles that bifurcate from a heteroclinic loop in a cubic non-elliptic Hamiltonian system by Melnikov functions of any order. The heteroclinic loop connects a non-Morse point and a hyperbolic saddle. We establish the asymptotic expansions of the Melnikov functions near this heteroclinic loop in near-Hamiltonian systems and provide the formulas of its first seven coefficients. As For a cubic degenerate Hamiltonian system, subject to $n$th-degree polynomial perturbations, we derive the precise number of limit cycles near the loop by using the first-order Melnikov function. When $n=3$, we obtain 3, 5, 6 and 1 limit cycle bifurcating from the heteroclinic loop by the first, second, third and forth order Melnikov functions, respectively.

报告人简介：孙宪波，杭州师范大学教授，理学博士，博士生导师。研究方向为微分方程定性理论及其应用，在J. Differential Equations、Discrete and Continuous Dynamical Systems-A等国际主流SCI期刊上发表学术论文30余篇，主持多项国家自然科学基金及省部级项目。