Bifurcations of the Riccati Quadratic Polynomial Differential Systems

Abstract

In this paper, we characterize the global phase portrait of the Riccati quadratic polynomial differential system (x) over dot = alpha(2) (x), (y) over dot = ky(2) + beta(1)(x)y + -gamma(2)(x), with (x,y) is an element of R-2, gamma(2)(x) nonzero (otherwise the system is a Bernoulli differential system), k not equal 0 (otherwise the system is a Lienard differential system), beta(1)(x) a polynomial of degree at most 1, alpha(2)(x) and -gamma(2)(x) polynomials of degree at most 2, and the maximum of the degrees of alpha(2)(x) and ky(2) + beta(1)(x)y + gamma(2)(x) is 2. We give the complete description of the phase portraits in the Poincare disk (i.e. in the compactification of R-2 adding the circle S-1 of the infinity) modulo topological equivalence.