GLOBAL PHASE PORTRAITS OF SOME REVERSIBLE CUBIC CENTERS WITH NONCOLLINEAR SINGULARITIES

Abstract

The results in this paper show that the cubic vector fields ẋ = -y + M(x, y) - y(x2+ y2), ẏ = x + N(x, y) + x( x2+ y2), 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)x2+ α2λy2with α, γ ∈ ℝ and λ ≠ 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 γλ < 0 and at least 46 for γλ > 0. Furthermore, the global bifurcation diagram is analyzed.