  We study the maximum number of limit cycles that can bifurcate from a
degenerate center of a cubic homogeneous polynomial differential system. Using
the averaging method of second order and perturbing inside the class of all
cubic polynomial differential systems we prove that at most three limit cycles
can bifurcate from the degenerate center. As far as we know this is the first
time that a complete study up to second order in the small parameter of the
perturbation is done for studying the limit cycles which bifurcate from the
periodic orbits surrounding a degenerate center (a center whose linear part is
identically zero) having neither a Hamiltonian first integral nor a rational
one. This study needs many computations, which have been verified with the help
of the algebraic manipulator Maple.
