Phase Portrait of the Riccati Quadratic Polynomial Differential Systems

Abstract

In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}={\alpha }_2(x),\dot{y}=k{y}^2+{\beta }_1(x)y+{\gamma }_2(x),$$ with $(x,y)\in {\mathbb{R}}^2$, ${\gamma }_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\ne 0$ (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 $k{y}^2+{\beta }_1(x)y+{\gamma }_2(x)$ is 2. We give the complete description of their phase portraits in the Poincare disk

arXiv comment22 pages

References in corpus (1)

• Riccati equations for bounded radiating systems