报告题目1：Normal forms of polynomial systems

报告人：Valery Romanovski Professor，University of Maribor

报告时间：2024年9月23日（周一）8:30-9:30

报告地点：20-308

报告摘要：There are two ways to compute Poincaré-Dulac normal forms of systems of ODEs. Under the original approach used by Poincare and Dulac the normalizing transformation is explicitly computed. On each step, the normalizing procedure requires the substitution of a polynomial to a series. Under the other approach, a normal form is computed using Lie transformations. In this case, the changes of coordinates are performed as actions of certain infinitesimal generators. In both cases, on each step the homological equation is solved in the vector space of polynomial vector fields where each component of the vector field is a homogeneous polynomial. We present the third way of computing normal forms of polynomial systems of ODEs where the coefficients of all terms are parameters. Although we use Lie transforms, the homological equation is solved not in the space of homogeneous vector fields of degree j, but in the vector space of polynomial vector fields where each component is a homogeneous polynomialin the parameters of the system. It is shown that the space of the parameters is a kind of dual space and the computation of normal forms can be performed in the space of parameters treated as the space of generalized vector fields, which we call the semilattice vector fields. The approach provides a simple way to parallelize the normal form computations opening the way to compute normal forms up to higher order than under previously known two approaches.

报告人简介：Valery Romanovski is a Professor at the Faculty of Electrical Engineering and Computer Science of University of Maribor and at the Faculty of Natural Science and Mathematics of University of Maribor. He is an excellent specialist in the research field of applying Computer Algebra to study the linearization, limit cycles and center-focus problem. He has received the Zoisovo priznanje (the state prize of second degree of the Republic of Slovenia).  Prof. Valery G. Romanovski, received his PhD in mathematics from Leningrad State University in 1986 and DSc in mathematics from the Institute of Mathematics of National Academy of Science of Belarus. He is a well-established scientist and the author/coauthor of about 100 research article in the field of ODEs and dynamical systems published in leading international journals.  He is also a co-author of the book which appears to be the first book which ties together dynamical systems and computational algebra. He has carried out joint studies with many researchers in Beijing, Shanghai, Chengdu and Zhejiang in China.

报告题目2：Blow-up method for proving integrability of some planar polynomial differential systems

报告人：Brigita Ferčec Assistant Professor，University of Maribor

报告时间：2024年9月23日（周一）9:30-10:30

报告地点：20-308

报告摘要：I shall discuss an effective method for proving integrability of the resonant saddles. The method is based on the use of a blow-up transformation and solving the recurrence differential equations using induction. Using this method some open integrability problems for certain resonant saddles are solved.

报告人简介：Brigita Ferčec is the Assistant Professor at the Faculty of Energy Technology of University of Maribor and at the Faculty of Natural Science and Mathematics of University of Maribor. She is also researcher at the Center for applied mathematics and theoretical physics. She finished her master degree in 2009 and her PhD in 2013 at University of Maribor. Her mentor for PhD was Prof. Valery Romanovskij and the topic was qualitative analysis of systems of ordinary differential equations. The fields of her research are differential equations and  dynamical systems.

报告题目3：Bifurcations of small limit cycles in some Liénard systems

报告人：田云教授，上海师范大学

报告时间：2024年9月23日（周一）10:30-11:30

报告地点：20-308

报告摘要：In this talk, we find Hopf cyclicity at the origin in some trigonometric Liénard systems, and obtain the maximum number of small limit cycles produced near a nilpotent singular point in some polynomial Liénard systems. To study the independence of focus values we find another family of equivalent quantities.

报告人简介：田云，上海师范大学数理学院教授，博士生导师，博士毕业于加拿大西安大略大学应用数学系，从事常微分方程定性理论、计算机符号计算和传染病模型等方向的研究，特别关注弱化的Hilbert第16问题、同宿异宿极限环分支和规范型的符号计算等相关问题。近年来，在JDE、Commun. Nonl. Sci. Numer. Simul和Nonlinear Anal. RWA等本领域主流期刊发表三十余篇学术论文。