Electron. J. Differential Equations, Vol. 2021 (2021), No. 69, pp. 1-52.

Quadratic systems with an invariant algebraic curve of degree 3 and a Darboux invariant

Jaume Llibre, Regilene D. S. Oliveira, Camila A. B. Rodrigues

Let QS be the class of non-degenerate planar quadratic differential systems and QS3 its subclass formed by the systems possessing an invariant cubic f(x,y)=0. In this article, using the action of the group of real affine transformations and time rescaling on QS, we obtain all the possible normal forms for the quadratic systems in QS3. Working with these normal forms we complete the characterization of the phase portraits in QS3 having a Darboux invariant of the form f(x, y)est, with s in R.

Submitted September 25, 2020. Published August 16, 2021.
Math Subject Classifications: 34C05, 34A34, 34C23.
Key Words: Quadratic vector fields; algebraic invariant curve; Darboux invariant; global phase portrait.

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Jaume Llibre
Departament de Matematiques
Universitat Aut7oacute;noma de Barcelona
08193 Bellaterra, Barcelona, Catalonia, Spain
email: jllibre@mat.uab.cat
Regilene D. S. Oliveira
Instituto de Ci\ências Matemáticas e Computação
Universidade de São Paulo
Avenida Trabalhador São-carlense, 400, 13.560-970
São Carlos, SP, Brazil
email: regilene@icmc.usp.br
Camila A. B. Rodrigues
Departamento de Matemática
Universidade Federal de Santa Catarina, 88040-900
Florianópolis, Santa Catarina, Brazil <
email: c.r.lima@ufsc.br

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