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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)

Data: 2014
Resum: 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.
Nota: Número d'acord de subvenció MINECO/MTM2008-03437
Nota: Número d'acord de subvenció AGAUR/2009/SGR-410
Nota: Agraïments: The first author is also supported by the grant AP2009-1189
Drets: Tots els drets reservats.
Llengua: Anglès
Document: article ; recerca ; preprint
Matèria: 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
Publicat a: Journal of Mathematical Analysis and Applications, Vol. 413 (2014) , p. 321-342, ISSN 0022-247X

DOI: 10.1016/j.jmaa.2013.11.047

27 p, 1007.0 KB

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