3-dimensional piecewise linear and quadratic vector fields with invariant spheres
We consider the class $\mathcal{X}$ of $3$-dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence o...
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| Format: | Article |
| Language: | English |
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University of Szeged
2024-08-01
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| Series: | Electronic Journal of Qualitative Theory of Differential Equations |
| Subjects: | |
| Online Access: | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11057 |
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| _version_ | 1846091042239021056 |
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| author | Claudio Buzzi Ana Livia Rodero Joan Torregrosa |
| author_facet | Claudio Buzzi Ana Livia Rodero Joan Torregrosa |
| author_sort | Claudio Buzzi |
| collection | DOAJ |
| description | We consider the class $\mathcal{X}$ of $3$-dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence of at least ten $1$-parameter families of crossing closed trajectories for quadratic vector fields in $\mathcal{X}$. |
| format | Article |
| id | doaj-art-ce4ae98b7dd14aa69ba4c99dda9f07d8 |
| institution | Kabale University |
| issn | 1417-3875 |
| language | English |
| publishDate | 2024-08-01 |
| publisher | University of Szeged |
| record_format | Article |
| series | Electronic Journal of Qualitative Theory of Differential Equations |
| spelling | doaj-art-ce4ae98b7dd14aa69ba4c99dda9f07d82025-01-15T21:24:58ZengUniversity of SzegedElectronic Journal of Qualitative Theory of Differential Equations1417-38752024-08-0120244312710.14232/ejqtde.2024.1.43110573-dimensional piecewise linear and quadratic vector fields with invariant spheresClaudio Buzzi0https://orcid.org/0000-0003-2037-8417Ana Livia RoderoJoan Torregrosa1Instituto de Biociências Letras e Ciências Exatas, Universidade Estadual Paulista, São José do Rio Preto, BrazilInstituto de Biociências Letras e Ciências Exatas, Universidade Estadual Paulista, São José do Rio Preto, Brazil & Departamento de Matemática, Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos, BrazilWe consider the class $\mathcal{X}$ of $3$-dimensional piecewise smooth vector fields that admit a first integral which leaves invariant any sphere centered at the origin. In this class, we prove that a linear vector field does not admit isolated invariant cones. Moreover, we provide the existence of at least ten $1$-parameter families of crossing closed trajectories for quadratic vector fields in $\mathcal{X}$.http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11057piecewise smooth vector fields with invariant spheresinvariant cones$1$-parameter families of closed trajectories |
| spellingShingle | Claudio Buzzi Ana Livia Rodero Joan Torregrosa 3-dimensional piecewise linear and quadratic vector fields with invariant spheres Electronic Journal of Qualitative Theory of Differential Equations piecewise smooth vector fields with invariant spheres invariant cones $1$-parameter families of closed trajectories |
| title | 3-dimensional piecewise linear and quadratic vector fields with invariant spheres |
| title_full | 3-dimensional piecewise linear and quadratic vector fields with invariant spheres |
| title_fullStr | 3-dimensional piecewise linear and quadratic vector fields with invariant spheres |
| title_full_unstemmed | 3-dimensional piecewise linear and quadratic vector fields with invariant spheres |
| title_short | 3-dimensional piecewise linear and quadratic vector fields with invariant spheres |
| title_sort | 3 dimensional piecewise linear and quadratic vector fields with invariant spheres |
| topic | piecewise smooth vector fields with invariant spheres invariant cones $1$-parameter families of closed trajectories |
| url | http://www.math.u-szeged.hu/ejqtde/periodica.html?periodica=1¶mtipus_ertek=publication¶m_ertek=11057 |
| work_keys_str_mv | AT claudiobuzzi 3dimensionalpiecewiselinearandquadraticvectorfieldswithinvariantspheres AT analiviarodero 3dimensionalpiecewiselinearandquadraticvectorfieldswithinvariantspheres AT joantorregrosa 3dimensionalpiecewiselinearandquadraticvectorfieldswithinvariantspheres |