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| In this work we extend well-known techniques for solving the Poincar\'e-Lyapunov nondegenerate analytic center problem in the plane to the 3-dimensional center problem at the analytic zero-Hopf singularity. We will see that both problems have much in common.
Thus we characterize the existence of a neighborhood of the zero-Hopf singularity completely foliated by periodic orbits (including continua of equilibriums) via an analytic Poincar\'e return map. We also prove that the 3-dimensional center is characterized by the fact that the system is analytically completely integrable and also because the Poicar\'e-Dulac normal form is analytic and orbitally linearizable.
Also we show that, when the system is polynomial and parametrized by its coefficients, the set of systems with 3-dimensional centers corresponds to an affine variety in the parameter space of coefficients. |
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