The center problem for Z2-symmetric nilpotent vector fields

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:466

Article

Algaba, Antonio¹  Garcia, Cristobal¹  Gine, Jaume²  Llibre, Jaume³ 

[1] Univ Huelva, Fac Ciencias, Dept Matemat, Huelva, Spain

[2] Univ Lleida, Inspires Res Ctr, Dept Matemat, Av Jaume II 69, Lleida 25001, Catalonia, Spain

[3] Univ Autonomous Barcelona, Dept Maternat, E-08193 Barcelona, Catalonia, Spain

关键词: Z(2)-symmetric differential systems;    Center problem;    Nilpotent singularity;   

DOI  :  10.1016/j.jmaa.2018.05.079

来源: Elsevier

【 摘 要 】

We say that a polynomial differential system x(over dot) = P(x,y), y(over dot) = Q(x,y) having the origin as a singular point is Z(2)-symmetric if P(-x, -y) = -P(x, y) and Q(-x, -y) = -Q(x, y). It is known that there are nilpotent centers having a local analytic first integral, and others which only have a C-infinity first integral. However these two kinds of nilpotent centers are not characterized for different families of differential systems. Here we prove that the origin of any Z(2)-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x, y) = y(2) + . . . , where the dots denote terms of degree higher than two. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

Free