Analytic Nilpotent Centers on Center Manifolds

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 $\lambda \ne 0$. 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.