Center Conditions for Nilpotent Singularities in the Plane Using Invariant Solutions
Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particula...
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| Tipo de recurso: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2024 |
| País: | España |
| Institución: | Varias* (Consorci de Biblioteques Universitáries de Catalunya, Centre de Serveis Científics i Acadèmics de Catalunya) |
| Repositorio: | Recercat. Dipósit de la Recerca de Catalunya |
| OAI Identifier: | oai:recercat.cat:10459.1/466317 |
| Acceso en línea: | https://doi.org/10.1007/s12346-024-01109-6 https://hdl.handle.net/10459.1/466317 |
| Access Level: | acceso abierto |
| Palabra clave: | Nilpotent center problem Analytic integrability Polynomial differential systems Cherkas method Takens normal form Decomposition in prime ideals |
| Sumario: | Recalling that at any regular point we always have a unique particular solution curve passing through it. In this work it is constructed such particular solution curve not passing through the nilpotent singularity but as close as we want to the singularity. By product the existence of such particular curve allows to use it to determine necessary conditions to have a center for nilpotent singularities in the plane. Several involve methods to solve the center problem are known all based in the existence of a change of variables and a scaling transformation of time bringing any differential system with a nilpotent center into a time-reversible system. Here we present a new algebraic method based on the existence of such particular solution curve not passing through the singular point and the involution associated to the nilpotent system with a center. The algebraic method needs the computation of this particular curve up to certain order, which can be done with the help of an algebraic manipulator. Finally a new algebraic method is derived computing the vanishing of a unique function which really gives a scalar method for computing the necessary conditions. |
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