Global classification of a class of cubic vector fields whose canonical regions are period annuli

We study cubic vector fields with inverse radial symmetry, i.e. of the form ẋ = δx - y + ax² + bxy + cy² + σ(dx - y)(x² + y²),ẏ = x + δy + ex² + fxy + gy² + σ(x + dy) (x² + y²), having a center at the origin and at infinity; we shortly call them cubic irs-systems. These systems are known to be Hamiltonian or reversible. Here we provide an improvement of the algorithm that characterizes these systems and we give a new normal form. Our main result is the systematic classification of the global phase portraits of the cubic Hamiltonian irs-systems respecting time (i.e. σ = 1) up to topological and diffeomorphic equivalence. In particular, there are 22 (resp. 14) topologically different global phase portraits for the Hamiltonian (resp. reversible Hamiltonian) irs-systems on the Poincaré disc. Finally we illustrate how to generalize our results to polynomial irs-systems of arbitrary degree. In particular, we study the bifurcation diagram of a 1-parameter subfamily of quintic Hamiltonian irs-systems. Moreover, we indicate how to construct a concrete reversible irs-system with a given configuration of singularities respecting their topological type and separatrix connections.