Analytic Nilpotent Centers on Center Manifolds
Claudio Pessoa, Lucas Queiroz

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 for some . 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 , 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.
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Taxonomy
TopicsAdvanced Differential Equations and Dynamical Systems · Advanced Topics in Algebra · Quantum chaos and dynamical systems
