Quadratic vector fields in class I
Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Chen, Hebai (Central South University. School of Mathematics and Statistics (China))
Ferrer, Lluc Manel (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Jia, Man (Central South University. School of Mathematics and Statistics (China))

Date:	2025
Abstract:	In [Ye et al. , Theory of Limit Cycles, 1986], quadratic systems are classified into three different normal forms (I, II and III) with increasing number of parameters. The simplest family is I and even several subfamilies of it have been studied, and some global attempts have been done, up to this paper, the full study was still undone. In this article, we make an interdisciplinary global study of Class I. Since the family has four parameters, we have studied it using the same technique that has already been used in several papers with similar systems which is based on the algebraic invariants of the Sibirskii's school. The bifurcation diagram for this class, done in the adequate parameter space which is the 3-dimensional real projective space, is quite rich in its complexity and yields 261 subsets with 49 different phase portraits for Class I (2 of them corresponding to linear systems), 7 of which have limit cycles. The phase portraits are always represented in the Poincaré disc. The bifurcation set is formed by an algebraic set of bifurcations of singularities, finite or infinite and by an analytic set of curves corresponding to phase portraits which have separatrix connections. Algebraic invariants were needed to construct the algebraic part of the bifurcation set, symbolic computations to deal with some quite complex invariants and numerical calculations to determine the position of the analytic bifurcation set of connections.
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, i la comunicació pública de l'obra, sempre que no sigui amb finalitats comercials, i sempre que es reconegui l'autoria de l'obra original. No es permet la creació d'obres derivades.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Quadratic vector fields ; Phase portraits ; Bifurcation diagram
Published in:	Dynamical Systems, Vol. 40, Issue 2 (2025) , p. 191-222, ISSN 1468-9375

DOI: 10.1080/14689367.2024.2436223

28 p, 687.3 KB

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