Classification of singular points of perturbed quadratic systems

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.

Classification of singular points of perturbed quadratic systems Keywords:

Quadratic system , Classification of singular points , Poincare index