Bifurcations of the Riccati Quadratic Polynomial Differential Systems

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system ? = a2(x), = ky2 + ß 1(x)y + ?2(x), with (x,y) 2, ?2(x) nonzero (otherwise the system is a Bernoulli differential system), k0 (otherwise the system is a Liénard differential system), ß1(x) a polynomial of degree at most 1, a2(x) and ?2(x) polynomials of degree at most 2, and the maximum of the degrees of a2(x) and ky2 + ß 1(x)y + ?2(x) is 2. We give the complete description of the phase portraits in the Poincaré disk (i.e. in the compactification of R2 adding the circle 1 of the infinity) modulo topological equivalence.