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Integrability of special quadratic systems with invariant hyperbolas

Part of: Smooth dynamical systems: general theory Qualitative theory

Published online by Cambridge University Press:  02 December 2024

Y Paulina Mancilla-Martínez Open the ORCID record for Y Paulina Mancilla-Martínez [Opens in a new window]  and
Claudia Valls Open the ORCID record for Claudia Valls [Opens in a new window]
Y Paulina Mancilla-Martínez
Affiliation:
Departamento de Matemática, Facultad de Ciencias, Universidad del Bío-Bío, Casilla 5-C, VIII-Región, Concepción, Chile ([email protected]) (corresponding author)
Claudia Valls
Affiliation:
Center for Mathematical Analysis, Geometry and Dynamical Systems, Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisboa, Portugal ([email protected])
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Abstract

In Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, the authors investigate about the integrability of the family QSH (the whole class of non-degenerate planar quadratic systems possessing at least one invariant hyperbola). However, some very difficult cases are left open in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90, and the main aim of this article is to study the Liouvillian integrability some of the systems that were left behind in Oliveira, Schlomiuk, Travaglini, and Valls, Geometry, integrability and bifurcation diagrams of a family of quadratic differential systems as application of Darboux theory of integrability, Electron. J. Qual. Theory Differ. Equ. 45(2021), 1–90.

Keywords

Darboux polynomialsinvariant algebraic curvesLiouvillian first integralspolynomial vector fieldsPuiseux series

MSC classification

Primary: 34C05: Location of integral curves, singular points, limit cycles
Secondary: 37C10: Vector fields, flows, ordinary differential equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , First View , pp. 1 - 27
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

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