Limit Cycles of a Quadratic System with Two Parallel Straight Line-Isoclines

Bifurcation of limit cycles in piecewise quadratic differential systems with an invariant straight line

We solve the center-focus problem in a class of piecewise quadratic polynomial differential systems with an invariant straight line. The separation curve is also a straight line which is not invariant. We provide families having at the origin a weak-foci of maximal order. In the continuous class, the cyclicity problem is also solved, being $3$ such maximal number. Moreover, for the discontinuous class but without sliding segment, we prove the existence of $7$ limit cycles of small amplitude.

Planar quadratic differential systems with invariant straight lines of the total multiplicity 4

In this article we consider the action of affine group and time rescaling on planar quadratic differential systems. We construct a system of representatives of the orbits of systems with four invariant lines, including the line at infinity and including multiplicities. For each orbit we exhibit its configuration. We characterize in terms of algebraic invariants and comitants and also geometrically, using divisors of the complex projective plane, the class of quadratic differential systems with four invariant lines. These conditions are such that no matter how a system may be presented, one can…

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。

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…

Limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center

We apply the averaging theory of high order for computing the limit cycles of discontinuous piecewise quadratic and cubic polynomial perturbations of a linear center. These discontinuous piecewise differential systems are formed by two either quadratic, or cubic polynomial differential systems separated by a straight line. We compute the maximum number of limit cycles of these discontinuous piecewise polynomial perturbations of the linear center, which can be obtained by using the averaging theory of order $n$ for $n=1,2,3,4,5$. Of course these limit cycles bifurcate from the periodic orbits o…

分段二次Kolmogorov系统中的极限环

Limit Cycles in Piecewise Quadratic Kolmogorov Systems

本文研究分段二次Kolmogorov系统，解决中心问题并分析穿越型极限环分岔，发现分段情形至少存在六个极限环，且最大弱焦点阶数不决定小振幅极限环的最大数量。