Abstract:

In this talk we provide with a complete topological classification of global phase portraits for the 6-parameter family of cubic vector fields of the form
begin{equation}dot{x}=-y+ax^2+bxy+cy^2-y(x^{2}+y^{2}),dot{y}=x+dx^2+exy+fy^2+x( x^{2}+y^{2}),end{equation}

that have simultaneously a center at the origin and at infinity. It is known that these vector fields are Hamiltoinian or reversible. In previous talks we focused on the classification of the Hamiltonian ones as well as the reversible ones with collinear singularities. In this talk we classify the remaining ones that are given by the 3-parameter family of reversible vector fields begin{equation}dot{x}=-y-gamma xy-y(x^{2}+y^{2}),dot{y}=x+( gamma-lambda) x^{2}+alpha^{2}lambda y^{2}+x( x^{2}+y^{2}),end{equation} where $alpha>0,gamma>0,lambdain mathbb{R}.$

The talk is based on a joint work with J. Torregrosa.