Global Phase Portraits Of Some Reversible Cubic Centers With Noncollinear Singularities

The results in this paper show that the cubic vector fields (x) over dot = - y vertical bar M(x, y) - y(x(2) vertical bar y(2)), (y) over dot = x + N(x, y) + x(x(2) + y(2)), 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) = -gamma xy, N(x, y) = (gamma - lambda) x(2) + a(2) lambda y(2) with alpha,gamma epsilon R and lambda = 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 five for gamma lambda < 0 and at least 46 for gamma lambda > 0. Furthermore, the global bifurcation diagram is analyzed.