Centers for a class of generalized quintic polynomial differential systems

Abstract

We classify the centers of the polynomial differential systems in R2 of degree d ≥ 5 odd that in complex notation writes as z˙ = iz + (zz¯)d−5/2 (Az5 + Bz4z¯ + Cz3z¯2 + Dz2z¯3 + Ezz¯4 + Fz¯5), where A, B, C, D, E, F ∈ C and either A = Re(D) = 0, or A = Im(D) = 0, or Re(A) = D = 0, or Im(A) = D = 0.

The first author is partially supported by a MINECO/ FEDER grant number MTM2011-22877 and an AGAUR (Generalitat de Catalunya) grant number 2014SGR 1204. The second author is partially supported by a MINECO/ FEDER grant number MTM2008-03437, an AGAUR grant number 2014SGR 568, ICREA Academia, two FP7-PEOPLE-2012-IRSES numbers 316338 and 318999, and FEDER-UNAB10-4E-378. The third author is supported by Portuguese National Funds through FCT – Fundação para a Ciência e a Tecnologia within the project PTDC/MAT/117106/2010 and by CAMGSD.