Bifurcation of the separatrix skeleton in some 1-parameter families of planar vector fields

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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.
Idioma originalEnglish
Pàgines (de-a)0989-1013
Nombre de pàgines25
RevistaJournal of Differential Equations
Volum259
DOIs
Estat de la publicacióAcceptat en premsa - 2015

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