Home > Articles > Published articles > Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields |
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 |
Rights: | Tots els drets reservats. |
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 |
Postprint 28 p, 932.5 KB |