A constructive approach to degenerate center problem

Abstract

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.