A constructive approach to degenerate center problem. (27th August 2022)

• Record Type: Journal Article
• Authors: MolaeiDerakhtenjani, M.; RabieiMotlagh, O.; MohammadiNejad, H.M.
• Is Part Of: International journal of dynamical systems and differential equations. Volume 12:Number 3(2022)
• Page Start: 247
• Page End: 266
• Publication Date: 2022-08-27
• Subjects: center problem -- degenerate center -- Poincare map -- polynomial systems -- Poincare coefficients -- system perturbations -- monodromicity -- symmetric systems -- Hamiltonian centers -- polar differential systems; Differential equations -- Periodicals; 515.35
• Languages: English
• ISSNs: 1752-3583

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.