Integrable zero-Hopf singularities and 3-dimensional centers

Abstract

In this paper we show that the well-known Poincaré-Lyapunov nondegenerate analytic center problem in the plane and its higher dimensional version expressed as the 3-dimensional center problem at the zero-Hopf singularity have a lot of common properties. In both cases the existence of a neighborhood of the singularity in the phase space completely foliated by periodic orbits (including equilibria) is characterized by the fact that the system is analytically completely integrable. Hence its Poincaré-Dulac normal form is analytically orbitally linearizable. There also exists an analytic Poincar\'e return map and, when the system is polynomial and parametrized by its coefficients, the set of systems with centers corresponds to an affine variety in the parameter space of coefficients. Some quadratic polynomial families are considered.

The author is partially supported by a MINECO grant number MTM2014-53703-P and by a CIRIT grant number 2014 SGR 1204.