TY - JOUR

T1 - The center problem for Z2-symmetric nilpotent vector fields

AU - Algaba, Antonio

AU - García, Cristóbal

AU - Giné, Jaume

AU - Llibre, Jaume

PY - 2018/10/1

Y1 - 2018/10/1

N2 - © 2018 Elsevier Inc. We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z2-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∞ 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 Z2-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x,y)=y2+⋯, where the dots denote terms of degree higher than two.

AB - © 2018 Elsevier Inc. We say that a polynomial differential system x˙=P(x,y), y˙=Q(x,y) having the origin as a singular point is Z2-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∞ 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 Z2-symmetric system is a nilpotent center if, and only if, there is a local analytic first integral of the form H(x,y)=y2+⋯, where the dots denote terms of degree higher than two.

KW - Center problem

KW - Nilpotent singularity

KW - Z -symmetric differential systems 2

U2 - https://doi.org/10.1016/j.jmaa.2018.05.079

DO - https://doi.org/10.1016/j.jmaa.2018.05.079

M3 - Article

VL - 466

SP - 183

EP - 198

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

ER -