10个小时的 希尔伯特16问题入门
[Mini Courses] Mini-Course: Qualitative theory of planar polynomial ODE
时间  Datetime
2023-10-17 21:00 — 23:00
地点  Venue
Zoom APP(2)()
报告人  Speaker
Professor Armengol Gasull
单位  Affiliation
Universitat Aut`onoma de Barcelona and CRM
邀请人  Host
肖冬梅
备注  remarks
From 21:00-23:30pm(in Beijing Time) on October 17 (Tuesday), October 19(Thursday),October 20(Friday),October 24(Tuesday),October 25(Wednesday) 加入 Zoom 会议 https://us06web.zoom.us/j/3518116587?pwd=ooaIjdFHE5YnmZo0nmW8dMxNBavWkW.1 会议号: 351 811 6587 密码: 202309
报告摘要  Abstract

The course is divided into ten one hour lessons. In the first two lessons we will recall some well-known tools and concepts appearing later. In particular, we will introduce the Hilbert 16th problem, the Abelian integrals, the Abel differential equations and we will study the stability of periodic orbits.

Lessons 3 and 4 are devoted to introduce some specific algebraic tools that are very useful in our approaches. In particular we will study and prove some results involving resultants, rational parameterizations and Poincar?e Miranda Theorem. All the results are illustrated with several examples of application.

In Lessons 5 and 6 we develop most of the presented results on existence and number of limit cycles of planar polynomial differential equations. The first lesson gives some results obtained the classical Lyapunov and Poincar?eBendixson approaches while Lesson 6 deals with a more new point of view. More specifically we study in a concrete family of planar ODE the probability of existence of limit cycle.

Lesson 7 formalizes the use of the harmonic balance method to obtain approximations of the limit cycles of a given system. In a few words this method approaches the limit cycles by their truncated Fourier series. We also illustrate how it can be used to study the period function.

In Lesson 8 we introduce a new point of view that looks the limit cycles as homoclinic solutions of a related system. In particular, this approach gives a tool to study the limit cycles that bifurcate from perturbations of reversible centers.

Lesson 9 shows how the qualitative and quantitative study of some planar differential equation can be used to know properties of some particular solutions of some partial differential equations: the so called traveling wave solutions. Finally, in Lesson 10 we introduce the so called Parrondo paradox for discrete and continuous dynamical systems. This paradox is similar to the original one that was formulated in game theory
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