Phase Portrait of the Riccati Quadratic Polynomial Differential Systems

TL;DR

This paper provides a comprehensive classification of the phase portraits of Riccati quadratic polynomial differential systems with specific polynomial degree constraints, within the Poincare disk.

Contribution

It offers a complete characterization of phase portraits for a class of Riccati systems with polynomial coefficients up to degree two, extending existing classifications.

Findings

Complete phase portrait descriptions in the Poincare disk.

Identification of conditions for different dynamical behaviors.

Classification of systems based on polynomial degree and parameters.

Abstract

In this paper we characterize the phase portrait of the Riccati quadratic polynomial differential systems $$\dot{x}= \alpha_2(x),\quad\dot{y} = ky^2+\beta_1(x) y + \gamma_2(x), $$ with $(x,y)\in\mathbb{R}^2$, $\gamma_2(x)$ non-zero (otherwise the system is a Bernoulli differential system), $k\neq0$ (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