一类平面等时中心的极限环分支  

Bifurcation of limit cycles of a class of planar isochronous centers

作　　者：魏怡倩 梁海华 WEI Yi-qian;LIANG Hai-hua(School of Mathematics and Systems Science,Guangdong Polytechnic Normal University,Guangzhou 510665,China)

机构地区：[1]广东技术师范大学数学与系统科学学院,广东广州510665

出　　处：《高校应用数学学报（A辑）》2021年第4期398-410,共13页Applied Mathematics A Journal of Chinese Universities(Ser.A)

基　　金：国家自然科学基金(11771101);广东省普通高校基础研究重大项目(2017KZDXM054);广州市科技计划项目(201805010001)。

摘　　要：考虑具有等时中心的2n+3次平面多项式微分系统x^(·)=-y+x^(2)y(x^(2)+y^(2))n;y^(·)=x+xy^(2)(x^(2)+y^(2))^(n):利用一阶平均法研究了该系统在m次多项式扰动下从中心的周期环域分支出来的极限环的最大个数(记为H(m)).首先证明了当n 2 N时,对任何正整数k均有H(2k¡1)=H(2k);其次,对n=1,利用Chebyshev判据,Sturm定理及定性分析技巧证明了H(1)=H(2)=1;H(3)=H(4)=4;H(5)=H(6)=8:该结果揭示了同一系统在不同次数的扰动下产生的极限环最大个数的某些规律.This paper considers the planar polynomial differential system of degree 2n+3 with isochronous center x^(·)=-y+x^(2)y(x^(2)+y^(2))n;y^(·)=x+xy^(2)(x^(2)+y^(2))^(n):By using the averaging theory of¯rst order,the maximum number of limit cycles that bifurcate from the periodic annulus of the isochronous center of the above system is studied.For anyxed n 2 N,when the system is perturbed inside the class of all polynomial differential systems of degree m,the maximum number of limit cycles,which is denoted by H(m),is proved to be satised that H(2k-1)=H(2k)(k∈Z_(+)).For n=1,the upper bounds H(1)=H(2)=1;H(3)=H(4)=4;H(5)=H(6)=8 are given by using Chebyshev's criterion,Sturm's theorem and the qualitative analysis technique.This result reveals some rules of the maximum numbers of limit cycles produced by the same system under different perturbations.

关 键 词：极限环 分支 平均法 等时中心 

分 类 号：O175[理学—数学]