Abstract

In this talk I report on a joint work with J. Llibre and J. Torregrosa. This work forms part of the classification of cubic
vector fields having a center both at the origin as at infinity. Here we classify the global phase portraits for the Hamiltonian vector fields in the 3-parameter family
[
X_{left(  g,c,e ight)  }leftrightarrowleft{
begin{array}
dot{x} & =-y-2gxy+cy^{2}-yleft(  x^{2}+y^{2} ight)  \
dot{y} & =x+ex^{2}+gy^{2}+xleft(  x^{2}+y^{2} ight)
end{array}
ight.  ,text{ }c,g,einmathbb{R},
]
with respect to topological and diffeomorphic equivalence. Attention goes to the systematic approach based on the number of singularities and the relative order of their Hamiltonian values. The techniques introduced in this approach are not restrictive to cubic Hamiltonian systems as we illustrate by constructing families of quintic (non-)Hamiltonian vector fields with a prescribed bifurcation diagram of phase portraits.