Quadratic perturbations of quadratic codimension-four centers
Quadratic perturbations of quadratic codimension-four centers
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DOI：
10.1016/j.jmaa.2009.04.004
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发表时间：
2008-11
影响因子：
1.3
通讯作者：
L. Gavrilov;I. Iliev
中科院分区：
文献类型：
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作者：
L. Gavrilov;I. Iliev
我们研究具有中心的所有二次微分系统x =P2（x，y），y =Q2（x，y）的集合中的层，称为余维四情况Q4。它有一个中心和一个节点，以及一个有理第一积分。系统在二次小扰动下的极限环由第一个Poincaré-Pontryagin-Melnikov积分I的零点决定。我们证明了未扰动系统的轨道是椭圆曲线，并且I是完全椭圆积分。然后使用Picard-Fuchs方程和Petrov方法（基于幅角原理），我们设置了一个上限为八个极限环的数量产生的周期环周围的中心。
We study the stratum in the set of all quadratic differential systems x˙=P2(x,y), y˙=Q2(x,y) with a center, known as the codimension-four case Q4. It has a center and a node and a rational first integral. The limit cycles under small quadratic perturbations in the system are determined by the zeros of the first Poincaré–Pontryagin–Melnikov integral I. We show that the orbits of the unperturbed system are elliptic curves, and I is a complete elliptic integral. Then using Picard–Fuchs equations and the Petrov's method (based on the argument principle), we set an upper bound of eight for the number of limit cycles produced from the period annulus around the center.