Invariant conditions for phase portraits of quadratic systems with complex conjugate invariant lines meeting at a finite point
Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Llibre, Jaume (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Schlomiuk, Dana (Université de Montréal. Département de Mathématiques et de Statistiques (France))
Vulpe, Nicolae (Vladimir Andrunakievichi Institute of Mathematics and Computer Science (Moldova))

Date:	2020
Abstract:	The goal of this article is to give invariant necessary and sufficient conditions for a quadratic system, presented in whatever normal form, to have anyone of 17 out of the 20 phase portraits of the family of quadratic systems with two complex conjugate invariant lines intersecting at a finite real point. The systems in this family have a maximum of one limit cycle. Among the 17 phase portraits we have two with limit cycles. We also give invariant necessary and sufficient conditions for a system to have one of the three remaining phase portraits, out of which one has a limit cycle and another one a homoclinic loop. In the region R determined by these last conditions, due to the presence of systems with a homoclinic loop, an analytic condition, the three phase portraits cannot be separated by algebraic conditions in terms of invariant polynomials. We also give the bifurcation diagram of this family, outside the region R, in the twelve parameter space of coefficients of the systems.
Grants:	Ministerio de Economía y Competitividad MTM2016-77278-P Agència de Gestió d'Ajuts Universitaris i de Recerca 2017/SGR-1617 European Commission 777911
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Quadratic vector fields ; Infinite and finite singularities ; Affine invariant polynomials ; Poincar compactification ; Topological configuration of singularities ; Phase portrait ; Limit cycle
Published in:	Rendiconti del Circolo Matematico di Palermo, vol. 70 (July 2020) p. 923-945, ISSN 1973-4409

DOI: 10.1007/s12215-020-00541-2

Postprint 25 p, 548.4 KB

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