Global phase portraits of some reversible cubic centers with noncollinear singularities
Caubergh, Magdalena (Universitat Autònoma de Barcelona. Departament de Matemàtiques)
Torregrosa, Joan (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2013
Abstract:	The results in this paper show that the cubic vector fields ˙x = -y + M(x, y) - y(x2 + y2), y˙ = x + N(x, y) + x(x2 + y2), where M, N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end the reversible subfamily defined by M(x, y) = -γxy, N(x, y) = (γ - λ)x2 + α2λy2 with α, γ ∈ R and λ 6= 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly 5 for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.
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 also is supported by the Ramón y Cajal grant number RYC-2011-07730.
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Reversible planar vector fields ; Cubic vector fields ; Global classification of phase portraits ; Bifurcation diagram
Published in:	International journal of bifurcation and chaos in applied sciences and engineering, Vol. 23 Núm. 9 (2013) , p. 1350161 (30 pages), ISSN 1793-6551

DOI: 10.1142/S0218127413501617

Postprint 38 p, 3.6 MB

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