Integrability and linearizability of a family of three-dimensional quadratic systems
We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of (1:-2:1) – resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpos...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2021 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/345608 |
| Acceso en línea: | https://hdl.handle.net/2117/345608 https://dx.doi.org/10.1080/14689367.2021.1893661 |
| Access Level: | acceso abierto |
| Palabra clave: | Hamilton-Jacobi equations Integrability Linearizability First integral Jacobi multiplier Darboux function Equacions de Hamilton-Jacobi Àrees temàtiques de la UPC::Matemàtiques i estadística::Equacions diferencials i integrals |
| Sumario: | We consider a three-dimensional vector field with quadratic nonlinearities and in general none of the axis plane is invariant. For our investigation, we are interesting in the case of (1:-2:1) – resonance at the origin. Hence, we deal with a nine parametric family of quadratic systems and our purpose is to understand the mechanisms of local integrability. By computing some obstructions, knowing as resonant focus quantities, first we present necessary conditions that guarantee the existence of two independent local first integrals at the origin. For this reason Gröbner basis and some other algorithms are employed. Then we examine the cases where the origin is linearizable. Some techniques like existence of invariant surfaces and Jacobi multipliers, Darboux method, properties of linearizable nodes of two dimensional systems and power series arguments are used to prove the sufficiency of the obtained conditions. For a particular three-parametric subfamily, we provide conditions on the parameters to guarantee the non-existence of a polynomial first integral. |
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