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arXiv · 2111.12047

Analytic Nilpotent Centers on Center Manifolds

Claudio Pessoa·Lucas Queiroz
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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.

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Claudio Pessoa, Lucas Queiroz. 2021-11-23. Analytic Nilpotent Centers on Center Manifolds. https://doi.org/10.1016/j.jmaa.2023.127120

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