First, doing a combination of analytical and algebraic computations, we determine by first time an explicit normal form depending only on three parameters for all cubic homogeneous polynomial differential systems having a center.
After using the averaging method of first order we show that we can obtain at most one limit cycle bifurcating from the periodic orbits of the mentioned centers when they are perturbed inside the class of all cubic polynomial differential systems. Moreover, there are examples with one limit cycles.
"Cubic homogeneous polynomial centers." Publ. Mat. 58 (S1) 297 - 308, 2014.