Classification of singular points of perturbed quadratic systems

Document Type : Research Paper


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


We consider the following two-dimensional differential system:
\[ \left\{\begin{array}{l}
\dot{x}=ax^{2}+bxy+cy^{2}+\Phi(x,y) \,, \\
\dot{y}=dx^{2}+exy+fy^{2}+\Psi(x,y) \,,
\end{array} \right.\]
in which $\lim_{(x,y)\rightarrow(0,0)}\frac{\Phi(x,y)}{x^{2}+y^{2}} = \lim_{(x,y)\rightarrow(0,0)}\frac{\Psi(x,y)}{x^{2}+y^{2}}=0$ and
$\Delta=(af-cd)^{2}-(ae-bd)(bf-ce)\neq0 $.
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.