Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields
Caubergh, Magdalena Maria (Universitat Autònoma de Barcelona. Departament de Matemàtiques)

Date:	2015
Abstract:	This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields X_m^k, defined by =y^3-x^2k 1,=-x my^4k 1, where m is a real parameter and k1 integer. The bifurcation diagram for the separatrix skeleton of X_m^k in function of m is determined and the one for the global phase portraits of (X^1_m)_mR is completed. Furthermore for arbitrary k1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of X_m^k is found to be uniformly bounded independent of mR and the Hilbert number for (X_m^k)_mR, that thus is finite, is found to be at least one.
Grants:	Ministerio de Economía y Competitividad MTM2008-03437 Ministerio de Economía y Competitividad MTM2013-40998P Agència de Gestió d'Ajuts Universitaris i de Recerca 2014/SGR-568
Note:	Agraïments: The author is supported by the Ramón y Cajal grant RYC-2011-07730
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Language:	Anglès
Document:	Article ; recerca ; Versió acceptada per publicar
Subject:	Global phase portrait ; Hilbert's 16th Problem ; Limit cycles ; Nilpotent center problem ; Rotated vector field ; Separatrix skeleton
Published in:	Journal of differential equations, Vol. 259 (2015) , p. 989-1013, ISSN 1090-2732

DOI: 10.1016/j.jde.2015.02.036

Postprint 28 p, 932.5 KB

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