Solution of the Center Problem for a Class of Polynomial Differential Systems  

作　　者：Chang Jian Liu Jaume Llibre Rafael Ramírez Valentín Ramírez 

机构地区：[1]School of Mathematics(Zhuhai),Sun Yat-sen University,Zhuhai,519086,P.R.China [2]Departament de Matemàtiques,Universitat Autònoma de Barcelona,08193,Bellaterra,Barcelona,Catalonia,Spain [3]Departament d’Enginyeria Informàtica i Matemàtiques,Universitat Rovira i Virgili,Avinguda dels Paiïsos Catalans 26,43007,Tarragona,Catalonia,Spain

出　　处：《Acta Mathematica Sinica,English Series》2024年第7期1685-1696,共12页数学学报（英文版）

基　　金：Supported by Grant NNSF of China(Grant No.12171491);the Ministerio de Ciencia,Innovación y Universidades,Agencia Estatal de Investigación grants MTM2016-77278-P(FEDER)and PID2019-104658GB-I00(FEDER);the Agència de Gestiód’Ajuts Universitaris i de Recerca grant 2017SGR1617;the H2020 European Research Council grant MSCA-RISE-2017-777911。

摘　　要：Consider the polynomial differential system of degree m of the form x=-y(1+μ(a_(2)x-a_(1)y))+x(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),y=x(1+μ(a_(2)x-a_(1)y))+y(v(a_(1)x+a_(2)y)+Ω_(m-1)(x,y)),whereμandνare real numbers such that(μ^(2)+v^(2))(μ+v(m-2))(a_(1)^(2)+a_(2)^(2))≠m>2 andΩ_(m−1)(x,y)is a homogenous polynomial of degree m−1.A conjecture,stated in J.Differential Equations 2019,suggests that whenν=1,this differential system has a weak center at the origin if and only if after a convenient linear change of variable(x,y)→(X,Y)the system is invariant under the transformation(X,Y,t)→(−X,Y,−t).For every degree m we prove the extension of this conjecture to any value ofνexcept for a finite set of values ofμ.

关 键 词：Center problem polynomial differential system 

分 类 号：O175[理学—数学]