New Exploration of Phase Portrait Classification of Quadratic Polynomial Differential Systems Based on Invariant Theory
Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Cairó, Laurent (Université d'Orleans)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2025
Abstract:	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.
Grants:	Agencia Estatal de Investigación PID2022-136613NB-I00 Agència de Gestió d'Ajuts Universitaris i de Recerca 2021/SGR-00113
Rights:	Aquest document està subjecte a una llicència d'ús Creative Commons. Es permet la reproducció total o parcial, la distribució, la comunicació pública de l'obra i la creació d'obres derivades, fins i tot amb finalitats comercials, sempre i quan es reconegui l'autoria de l'obra original.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Quadratic vector field ; Quadratic system ; Phase portrait
Published in:	AppliedMath, Vol. 5, Núm. 2 (June 2025) , art. 68, ISSN 2673-9909

DOI: 10.3390/appliedmath5020068

22 p, 563.7 KB

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