Document Type : Research Paper

Authors

• Asadollah Aghajani
• Mohsen Mirafzal

School of Mathematics, Iran University of Science and Technology, Narmak, Tehran 16844-13114, Iran

Abstract

We consider the following two-dimensional differential system:
$$\{\begin{array}{l}\dot{x}=a{x}^2+bxy+c{y}^2+\mathrm{\Phi }(x,y),\\ \dot{y}=d{x}^2+exy+f{y}^2+\mathrm{\Psi }(x,y),\end{array}$$
in which $\underset{(x,y)\to (0,0)}{lim}\frac{\mathrm{\Phi }(x,y)}{x^2+y^2}=\underset{(x,y)\to (0,0)}{lim}\frac{\mathrm{\Psi }(x,y)}{x^2+y^2}=0$ and $\mathrm{\Delta }=(af-cd{)}^2-(ae-bd)(bf-ce)\ne 0$. By calculating Poincare index and using Bendixson formula we will find all the possibilities under definite conditions for classifying the system by means of kinds of sectors around the origin which is an equilibrium point of degree two.

Keywords

• Quadratic system
• Classification of singular points
• Poincare index