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

On the number of limit cycles for polycycles $S^{(2)}$ and $S^{(3)}$ in quadratic Hamilton systems under perturbations of piecewise smooth polynomials

In this paper, by using Picard-Fuchs equations and Chebyshev criterion, we study the bifurcate of limit cycles for quadratic Hamilton system $S^{(2)}$ and $S^{(3)}$: $\dot{x}= y+2axy+by^2$, $\dot{y}=-x+x^2-ay^2$ with $a\in(-\frac{1}{2},1)$, $b=(1-a)(1+2a)^{1/2}$ and $a=1$, $b=0$ respectively, under perturbations of piecewise smooth polynomials with degree $n$. The discontinuity is the line $y=0$. We bound the number of zeros of first order Melnikov function which controls the number of limit cycles bifurcating from the center. It is proved that the upper bounds of the number of limit cycles fo…

分段平面二次微分系统中极限环的数量

On the number of limit cycles in piecewise planar quadratic differential systems

本文利用位移映射展开研究直线分区的分段平面二次系统，对 Loud 等时中心及非光滑中心进行一阶、二阶扰动分析，给出至少12个极限环的例子并纠正文献错误。

On the Hilbert number for piecewise linear vector fields with algebraic discontinuity set

The second part of the Hilbert's sixteenth problem consists in determining the upper bound $\mathcal{H}(n)$ for the number of limit cycles that planar polynomial vector fields of degree $n$ can have. For $n\geq2$, it is still unknown whether $\mathcal{H}(n)$ is finite or not. The main achievements obtained so far establish lower bounds for $\mathcal{H}(n)$. Regarding asymptotic behavior, the best result says that $\mathcal{H}(n)$ grows as fast as $n^2\log(n)$. Better lower bounds for small values of $n$ are known in the research literature. In the recent paper "Some open problems in low dimens…

Hilbert's 16th Problem for Quadratic Systems. New Methods Based on a Transformation to the Lienard Equation

Fractionally-quadratic transformations which reduce any two-dimensional quadratic system to the special Lienard equation are introduced. Existence criteria of cycles are obtained.

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".

平面分段多项式向量场中交叉极限环数量的估计

Growth estimate for the number of crossing limit cycles in planar piecewise polynomial vector fields

本文研究了平面分段多项式向量场中交叉极限环数量的增长问题，证明其至少以n²/4的速度增长，并改进了哈密顿向量场类的线性增长估计。

基于坐标变换TS模型与分段二次Lyapunov函数的非线性系统吸引域估计

Estimation of Regions of Attraction for Nonlinear Systems via Coordinate-Transformed TS Models and Piecewise Quadratic Lyapunov Functions

本文提出通过多重坐标变换生成多个TS模型表示，并结合分段二次Lyapunov函数分别计算吸引域后取并集，从而显著扩大非线性系统吸引域估计。

Crossing limit cycles of planar discontinuous piecewise differential systems formed by isochronous centers

These last years an increasing interest appeared for studying the planar discontinuous piecewise differential systems motivated by the rich applications in modelling real phenomena. One of the difficulties for understanding the dynamics of these systems is the study their limit cycles. In this paper we study the maximum number of crossing limit cycles of some classes of planar discontinuous piecewise differential systems separated by a straight line, and formed by combinations of linear centers (consequently isochronous) and cubic isochronous centers with homogeneous nonlinearities. For these …

具有非正则线分隔的幂零鞍点不连续分段微分系统的穿越极限环

Crossing limit cycles of discontinuous piecewise differential systems with nilpotent saddles separated by a nonregular line

研究由非正则切换线分隔的平面不连续分段微分系统的穿越极限环，其中子系统为线性微分中心和六类哈密顿幂零鞍系统之一，证明最多存在七个穿越极限环，并给出每个族具有四个穿越极限环的显式例子。

环面上具有非正则切换流形的分段光滑向量场的极限环

Limit cycles of piecewise smooth vector fields on torus with non-regular switching manifold

本文研究二维环面上分段光滑向量场的交叉极限环最大数量，基于n次多项式首次积分假设，运用代数曲线相交方法获得精确上界，并构造实例证明其最优性。

平面分段系统中多重交叉极限环的出现：具有等时中心和非光滑切换流形

Emergence of Multiple Crossing Limit Cycles in Planar Piecewise Systems with Isochronous Centers and Nonsmooth Switching Manifolds

研究平面分段系统中多重交叉极限环的产生机制，通过非光滑切换流形增强动力学复杂性，推导出交叉极限环数量的上界并构造实例。

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。

具有圆状切换流形的分段光滑系统中的极限环

Limit cycles in piecewise smooth systems with circular switching manifold

研究分段光滑系统中极限环的性质，利用莫比乌斯变换建立圆状与直线状不连续性的等价性，通过二阶平均和李雅普诺夫量得出极限环数量的下界，并证明分段反全纯系统中极限环的上界，同时构造了显式代数极限环。