New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory
After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided int...
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MDPI AG
2025-06-01
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| author | Joan Carles Artés Laurent Cairó Jaume Llibre |
| author_facet | Joan Carles Artés Laurent Cairó Jaume Llibre |
| author_sort | Joan Carles Artés |
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| description | After linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal. |
| format | Article |
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| institution | Kabale University |
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| language | English |
| publishDate | 2025-06-01 |
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| spelling | doaj-art-3cc4a7d7008349f2bd85b13a1360e7c22025-08-20T03:32:27ZengMDPI AGAppliedMath2673-99092025-06-01526810.3390/appliedmath5020068New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant TheoryJoan Carles Artés0Laurent Cairó1Jaume Llibre2Departament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, SpainInstitut Denis Poisson, Université d’Orleans, Collegium Sciences et Techniques, Batiment de Mathématiques, rue de Chartres BP6759, CEDEX 2, 45067 Orléans, FranceDepartament de Matemàtiques, Universitat Autònoma de Barcelona, Bellaterra, 08193 Barcelona, Catalonia, SpainAfter linear differential systems in the plane, the easiest systems are quadratic polynomial differential systems in the plane. Due to their nonlinearity and their many applications, these systems have been studied by many authors. Such quadratic polynomial differential systems have been divided into ten families. Here, for two of these families, we classify all topologically distinct phase portraits in the Poincaré disc. These two families have already been studied previously, but several mistakes made there are repaired here thanks to the use of a more powerful technique. This new technique uses the invariant theory developed by the Sibirskii School, applied to differential systems, which allows to determine all the algebraic bifurcations in a relatively easy way. Even though the goal of obtaining all the phase portraits of quadratic systems for each of the ten families is not achievable using only this method, the coordination of different approaches may help us reach this goal.https://www.mdpi.com/2673-9909/5/2/68quadratic vector fieldquadratic systemphase portrait |
| spellingShingle | Joan Carles Artés Laurent Cairó Jaume Llibre New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory AppliedMath quadratic vector field quadratic system phase portrait |
| title | New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory |
| title_full | New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory |
| title_fullStr | New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory |
| title_full_unstemmed | New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory |
| title_short | New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory |
| title_sort | new exploration of phase portrait classification of quadratic polynomial differential systems based on invariant theory |
| topic | quadratic vector field quadratic system phase portrait |
| url | https://www.mdpi.com/2673-9909/5/2/68 |
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