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

This article deals with the bifurcation of polycycles and limit cycles within the 1-parameter families of planar vector fields Xmk, defined by x˙=y3−x2k+1,y˙=−x+my4k+1, where m is a real parameter and k≥1 is an integer. The bifurcation diagram for the separatrix skeleton of Xmk in function of m is determined and the one for the global phase portraits of (Xm1)m∈R is completed. Furthermore for arbitrary k≥1 some bifurcation and finiteness problems of periodic orbits are solved. Among others, the number of periodic orbits of Xmk is found to be uniformly bounded independently of m∈R and the Hilbert number for (Xmk)m∈R, that thus is finite, is found to be at least one.