报告题目Ⅰ: Integrability and limit cycles in polynomial systems of ODE's

报 告 人：Valery Romanovski, CAMTP and University of Maribor

时间：2024年9月14日, 14:00—15:00

地点：数学楼301

报告摘要：We discuss two problems related to the theory of polynomial plane differential systems, that is, systems of the form \begin{equation} \label{1}\frac{dx}{dt}=P_{n}(x,y),\frac{dy}{dt}=Q_{n}(x,y),\end{equation} where $P_{n}(x,y), Q_{n}(x,y)$ are polynomials of degree $n$, $x$ and $y$ are real unknown functions.The first one is the problem of local integrability, that is, the problem of  finding local analytic integrals in a neighborhood of singular points of system (1). We present a computational approach to find integrable systems within given parametric families of systems and describe some mechanisms of integrability. The second problem is called the cyclicity problem, or the local 16th Hilbert problem, and is related to the estimation of the number of limit cycles arising in system (1) after perturbationsof integrable systems. The approach is algorithmic and is based on algorithms of computational commutative algebra relying on  the Groebner bases theory.

报告题目Ⅱ: Invariants and reversibility in polynomial systems of ODEs

报 告 人：Mateja Grašič，University of Maribor

时间：2024年9月14日, 15:00—16:00

地点：数学楼301

报告摘要：We first investigate the interconnection of invariants of certain group actions and time-reversibility of a class of two-dimensional polynomial systems with 1:-1 resonant singularity at the origin. The time-reversibility is related to the Sibirsky subvariety of the center (integrability) variety and it is known that every time-reversible system has a local analytic first integral  at the origin.  We propose a new algorithm to obtain a generating set for the Sibirsky ideal of such polynomial  systems and investigate some algebraic properties of this ideal. Then, we discuss a generalization of the concept of time-reversibility in the $n$-dimensional case considering the systems with $1:\zeta:\dots :\zeta^{n-1}$ resonant singularity at the origin (where $\zeta$ is a primitive $n$-th root of unity) and study a connection of such reversibility with the invariants of some group actions in the space of parameters of the system and Lawrence ideals.

报告人简介：

Valery Romanovski 现任斯洛文尼亚马里博尔(Maribor)大学教授, 应用数学与理论物理中心研究员，上海师范大学特聘教授。 于1986年列宁格勒州立大学(现圣彼得堡国立大学)获得物理与数学科学专业哲学博士，2001年白俄罗斯国家科学院数学研究所获物理与数学科学专业博士，主要研究方向为微分方程。Valery Romanovski教授2011年获斯洛文尼亚科学研究杰出贡献奖， 主持并参与多项国际科研项目，如2017斯洛文尼亚-匈牙利“微分方程中代数方法的应用”项目任首席研究员；2004、2006、2009、2012、2015斯洛文尼亚-美国双边项目任首席研究员；2008、20010、2011、2016斯洛文尼亚-俄罗斯双边项目任首席研究员等，并担任“Qualitative Theory of Dynamical Systems”、“Journal of Applied Analysis and Computation”等SCI杂志的编委，多次担任国际会议和讲习班的主讲人，组织过多次国际会议和讲习班。

Mateja Grašič 现任斯洛文尼亚马里博尔(Maribor)大学助理教授，于马里博尔获得理学博士。