On the Limit Cycles of a Class of Cubic Polynomial Differential Systems

One of the main problems studying the dynamics of a polynomial differential system in the plane is to determine the non-existence, existence and its number of limit cycles. Here we study the maximum number of limit cycles that bifurcate from the periodic solutions of the family of cubic polynomial centers ẋ=x ( x²-y²+2 x+α ²+1) , ẏ=y ( x²-y²-2 y-α ²-1) , x ˙ = x x 2 - y 2 + 2 x + α 2 + 1 , y ˙ = y x 2 - y 2 - 2 y - α 2 - 1 , where α ≠ 0 α ≠ 0 , when it is perturbed inside the classes of all cubic polynomial differential systems. We obtain by using the averaging method of first order that the maximum number of limit cycles is 3 and it is reached. The fact that the unperturbed center has the two invariant straight lines $$x=0$$ x = 0 and $$y=0$$ y = 0 can be the reason that the number of limit cycles of the perturbed systems be so small respect other cubic polynomial differential systems.