Title: A constructive approach to degenerate center problem

Authors: M. MolaeiDerakhtenjani; O. RabieiMotlagh; H.M. MohammadiNejad

Addresses: Department of Applied Mathematics, University of Birjand, P.O. Box 97175/615, Birjand, 9717434765, Iran ' Department of Applied Mathematics, University of Birjand, P.O. Box 97175/615, Birjand, 9717434765, Iran ' Department of Applied Mathematics, University of Birjand, P.O. Box 97175/615, Birjand, 9717434765, Iran

Abstract: We give a constructive approach to the degenerate center problem. First, we consider homogeneous polynomial systems and provide various conditions for which the origin is a center. Then, by using the Poincare coefficients in polar coordinate, we complete a rigorous computation such that the nonhomogeneous system perturbed by lower terms has an annular region surrounding the origin. This enables us to show that a degenerate center may be the limit of a linear center, a nilpotent singularity, and even a hyperbolic saddle point. Finally, we provide sufficient conditions such that the origin is a degenerate center for a nonhomogeneous system. The system may be of even degree, so we have degenerate centers of even degree, which are rare.

Keywords: center problem; degenerate center; Poincare map; polynomial systems; Poincare coefficients; system perturbations; monodromicity; symmetric systems; Hamiltonian centers; polar differential systems.

DOI: 10.1504/IJDSDE.2022.125204

International Journal of Dynamical Systems and Differential Equations, 2022 Vol.12 No.3, pp.247 - 266

Received: 29 Sep 2019
Accepted: 23 Oct 2020
Published online: 02 Sep 2022 *

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