Zero-Hopf polynomial centers of third-order differential equations

Abstract

We study the 3-dimensional center problem at the zero-Hopf singularity in some families of polynomial vector fields arising from third-order polynomial differential equations. After proving some general properties we check that the quadratic family has no 3-dimensional centers. Later we characterize all the 3-dimensional centers in the cubic homogeneous family. Finally we give a partial classification of the 3-dimensional centers at just one singularity of the full cubic family and propose one open problem to close this classification.

The first author is partially supported by a MINECO grant number MTM2014-53703-P and an AGAUR grant number 2014SGR 1204. The second author is supported by Portuguese National Funds through FCT - Fundaçao para a Ciencia e a Tecnologia within CAMGSD and the project PTDC/MAT/117106/2010.