Nilpotent Centers in $\mathbb{R}^3$

TL;DR

This paper investigates nilpotent singular points in three-dimensional differential systems, focusing on formal integrability and the center problem on the Center Manifold without polynomial approximations, with applications to specific dynamical systems.

Contribution

It introduces a novel approach to the center problem for nilpotent singularities that avoids polynomial approximations and applies to complex systems like the Lorenz and dynamo models.

Findings

Solved the Nilpotent Center Problem for the Generalized Lorenz system.

Solved the Nilpotent Center Problem for the Hide-Skeldon-Acheson dynamo system.

Provided new results for planar systems with nilpotent singularities.

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 study the formal integrability and the center problem for those types of singular points in the monodromic case. Our approach do not require polynomial approximations of the Center Manifold in order to study the center problem. As a byproduct, we obtain some useful results for planar $C^r$ systems having a nilpotent singularity. We conclude the work solving the Nilpotent Center Problem for the Generalized Lorenz system and the Hide-Skeldon-Acheson dynamo system.