Bifurcation diagram and stability for a one-parameter family of planar vector fields
García-Saldaña, Johanna Denise (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Gasull, Armengol (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Giacomini, Hector (Université de Tours(France). Laboratoire de Mathématiques et Physique Théorique)

Date:	2014
Abstract:	We consider the 1-parameter family of planar quintic systems, ˙x = y3−x3, y˙ = −x + my5, introduced by A. Bacciotti in 1985. It is known that it has at most one limit cycle and that it can exist only when the parameter m is in (0. 36, 0. 6). In this paper, using the Bendixon-Dulac theorem, we give a new unified proof of all the previous results, we shrink this to (0. 547, 0. 6), and we prove the hyperbolicity of the limit cycle. We also consider the question of the existence of polycycles. The main interest and difficulty for studying this family is that it is not a semi-complete family of rotated vector fields. When the system has a limit cycle, we also determine explicit lower bounds of the basin of attraction of the origin. Finally we answer an open question about the change of stability of the origin for an extension of the above systems.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Agència de Gestió d'Ajuts Universitaris i de Recerca 2009/SGR-410
Note:	Agraïments: The first author is also supported by the grant AP2009-1189
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Planar polynomial system ; Uniqueness and hyperbolicity of the limit cycle ; Polycycle ; Bifurcation ; Phase portrait on the Poincaré disc ; Dulac function ; Stability ; Nilpotent point ; Basin of attraction
Published in:	Journal of mathematical analysis and applications, Vol. 413 (2014) , p. 321-342, ISSN 1096-0813

DOI: 10.1016/j.jmaa.2013.11.047

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