Formal integrability for monodromic nilpotent singular points in R3

Abstract

Consider analytic 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 study the formal and analytic integrability for those types of singular points in the monodromic case. As a byproduct, we obtain some useful results for planar $C^r$ systems having a monodromic nilpotent singularity. We conclude the work by studying issues related to monodromy and formal integrability for the Elsonbaty–El-Sayed system, the Hide–Skeldon–Acheson dynamo system and the Generalized Lorenz system. For this last system, we were able to detect nilpotent centers.