An Equilibrium Point Analysis of a Class of Planar Cubic Polynomial Systems

The equilibrium points of a class of plane cubic polynomial systems ẋ=-y+αx²-αy²+βx³-3βxy², ẏ=x-2αxy+3βx²y-βy³ are discussed. It is proved that when |α-1|≪0, |β-1|≪0, there are four infinite equilibrium points and all of them are saddle points, and there are three finite equilibrium points and all of them are focal points. The position, order and stability of the three focal points are given.