Analytic Nilpotent Centers on Center Manifolds

TL;DR

This paper investigates nilpotent centers in three-dimensional analytic systems with a focus on center manifolds, establishing conditions for the existence of inverse Jacobi multipliers and connecting nilpotent centers to Hopf-type centers.

Contribution

It proves the existence of formal inverse Jacobi multipliers for systems with nilpotent centers of Andreev number 2 and relates nilpotent centers to Hopf-type centers in three dimensions.

Findings

Systems with nilpotent centers admit inverse Jacobi multipliers.

Nilpotent centers are limits of Hopf-type centers.

Results apply without explicit center manifold parametrization.

Abstract

Consider analytical three-dimensional differential systems having a singular point at the origin such that its linear part is $y\partial_x-\lambda z\partial_z$ for some $\lambda\neq 0$. The restriction of such systems to a Center Manifold has a nilpotent singular point at the origin. We prove that if the restricted system has an analytic nilpotent center at the origin, with Andreev number $2$, then the three-dimensional system admits a formal inverse Jacobi multiplier. We also prove that nilpotent centers of three-dimensional systems, on analytic center manifolds, are limits of Hopf-type centers. We use these results to solve the center problem for some three-dimensional systems without restricting the system to a parametrization of the center manifold.