Limit cycles of polynomial differential equations with quintic homogeneous nonlinearities

In this paper we mainly study the number of limit cycles which can bifurcate from the periodic orbits of the two centers x ̇ = − y , y ̇ = x ; x ̇ = − y ( 1 − ( x 2 + y 2 ) 2 ) , y ̇ = x ( 1 − ( x 2 + y 2 ) 2 ) ; when they are perturbed inside the class of all polynomial differential systems with quintic homogeneous nonlinearities. We do this study using the averaging theory of first, second and third orders.