Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

On quadratic orbital networks

These are some informal remarks on quadratic orbital networks over finite fields. We discuss connectivity, Euler characteristic, number of cliques, planarity, diameter and inductive dimension. We find a non-trivial disconnected graph for d=3. We prove that for d=1 generators, the Euler characteristic is always non-negative and for d=2 and large enough p the Euler characteristic is negative. While for d=1, all networks are planar, we suspect that for d larger or equal to 2 and large enough prime p, all networks are non-planar. As a consequence on bounds for the number of complete sub graphs of …

对空间中平面中心构型的海森矩阵：分解、莫尔斯指数与对称性约化

The Hessian of Planar Central Configurations in Pair Space: Decomposition, Morse Index and Symmetry Reduction

该研究推导对空间中平面中心构型的海森矩阵分解式，证明四天体中心构型莫尔斯指数至多为2，通过广义特征值表征海森矩阵性质，利用对称性约化证明菱形中心构型莫尔斯指数为0。

拟齐次多项式平面系统的中心

Centers of quasi-homogeneous polynomial planar systems

本文确定了0至4次拟齐次多项式平面向量场的中心，研究了各中心的可逆性与解析可积性，并发现了既非轨道可逆也非解析可积的多项式中心这一新场景。

Bifurcation of limit cycles from a switched equilibrium in planar switched systems and its application to power converters

We consider a switched system of two subsystems that are activated as the trajectory enters the regions $\{(x,y):x>\bar x\}$ and $\{(x,y):x<-\bar x\}$ respectively, where $\bar x$ is a positive parameter. We prove that a regular asymptotically stable equilibrium of the associated Filippov equation of sliding motion (corresponding to $\bar x=0$) yields an orbitally stable limit cycle for all $\bar x>0$ sufficiently small. The research is motivated by an application to a dc-dc power converter, where $\bar x>0$ is used in place of $\bar x=0$ to avoid sliding motions.