Global phase portraits of quadratic polynomial differential systems with a semi-elemental triple node
Artés Ferragud, Joan Carles (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Rezende, Alex C. (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2013
Abstract:	Planar quadratic differential systems occur in many areas of applied mathematics. Although more than one thousand papers have been written on these systems, a complete understanding of this family is still missing. Classical problems, and in particular, Hilbert's 16th problem [Hilbert, 1900, Hilbert, 1902], are still open for this family. In this article we make a global study of the family QT N of all real quadratic polynomial differential systems which have a semi-elemental triple node (triple node with exactly one zero eigenvalue). This family modulo the action of the affine group and time homotheties is three-dimensional and we give its bifurcation diagram with respect to a normal form, in the three-dimensional real space of the parameters of this form. This bifurcation diagram yields 28 phase portraits for systems in QT N counting phase portraits with and without limit cycles. Algebraic invariants are used to construct the bifurcation set. The phase portraits are represented on the Poincaré disk. The bifurcation set is not only algebraic due to the presence of a surface found numerically. All points in this surface correspond to connections of separatrices.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2005/SGR-550 European Commission 316338
Note:	Agraïments: The second author is supported by CAPES/DGU grant number BEX 9439/12-9 and the last author is partially supported by CAPES/DGU grant number 222/2010
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Published in:	International journal of bifurcation and chaos in applied sciences and engineering, Vol. 23 Núm. 8 (2013) , p. 1350140 (21 pages), ISSN 1793-6551

DOI: 10.1142/S021812741350140X

Postprint 22 p, 552.4 KB

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