希尔伯特的第16个二次系统问题基于Lienard方程变换的新方法

Lienard型方程的等距中心及其应用

Isochronous Centers of Lienard Type Equations and Applications

本文研究了中心为0的方程$(E) \ddot x + f(x) \dot x^2 + g(x) = 0$，并研究了它的等时性条件。当$f$和$g$是解析的（不一定是奇的）时，给出了0的等时性的一个充要条件.这种方法使我们能够提出一种算法，获得的条件（E）的一个点是一个等值线中心。特别地，我们又用另一种方法找到了二次Loud系统$(L_{D,F})$的等时线.也考虑了一些Kukles类。此外，我们还对一类具有等熵中心的5参数可逆三次系统进行了分类。关键词：周期函数，单调性，等时性，中心，多项式系统。

A restricted version of the Hilbert's 16th problem for quadratic vector fields

The restricted version of the Hilbert 16th problem for quadratic vector fields requires an upper estimate of the number of limit cycles through a vector parameter that characterizes the vector fields considered and the limit cycles to be counted. In this paper we give an upper estimate of the number of limit cycles of quadratic vector fields $"σ$--distant from centers and $\ka $-distant from singular quadratic vector fields" provided that the limit cycles are $"δ$--distant from singular points and infinity".

New lower bound for the Hilbert number in piecewise quadratic differential systems

We study the number of limit cycles bifurcating from a piecewise quadratic system. All the differential systems considered are piecewise in two zones separated by a straight line. We prove the existence of 16 crossing limit cycles in this class of systems. If we denote by $H_p(n)$ the extension of the Hilbert number to degree $n$ piecewise polynomial differential systems, then $H_p(2)\geq 16.$ As fas as we are concerned, this is the best lower bound for the quadratic class. Moreover, all the limit cycles appear in one nest bifurcating from the period annulus of some isochronous quadratic cente…

关于希尔伯特第16问题的一个变体

On a variant of Hilbert's 16th problem

本文研究平面多项式向量场极限环数目与单项式个数的关系，证明其至少二次增长，并给出$m\leqslant10$时的良好下界。

Hilbert 空间中的线性二次平均场博弈

Linear-Quadratic Mean Field Games in Hilbert spaces

本文首次在无限维 Hilbert 空间中发展线性二次平均场博弈理论，通过均值依赖将系统简化为 Riccati 方程与抽象发展方程的耦合系统，利用逼近程序证明解的存在唯一性，并应用于带延迟的生产规划问题。

关于近期尝试解决希尔伯特第16问题第二部分的一个注记

A note on a recent attempt to solve the second part of Hilbert's 16th Problem

本文通过指出近期提出的希尔伯特数公式$H(n)$为二次多项式，与已知的$n^2 \log n$渐近增长相矛盾，并结合反例，反驳了某研究声称已解决希尔伯特第16问题第二部分的观点。