Global phase Portraits of some Reversible cubic Centers with Noncollinear singularities

The results in this paper show that the cubic vector fields ẋ = −y + M(x, y) − y(x + y), ẏ = x + N(x, y) + x(x + y), where M,N are quadratic homogeneous polynomials, having simultaneously a center at the origin and at infinity, have at least 61 and at most 68 topologically different phase portraits. To this end the reversible subfamily defined by M(x, y) = −γxy, N(x, y) = (γ − λ)x + αλy with α, γ ∈ R and λ 6= 0, is studied in detail and it is shown to have at least 48 and at most 55 topologically different phase portraits. In particular, there are exactly 5 for γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.