Center problem in the center manifold for quadratic and cubic differential systems in R3R3

We obtain necessary and sufficient conditions for the existence of a center on a local center manifold for three six-parameter families of quadratic systems on R3R3. We also give a positive answer to the conjecture posed in Mahdi (2013) for a special class of systems, called the Moon–Rand systems in the particular case when λ=2λ=2 and f is a homogeneous cubic polynomial. We also solve the center problem for a natural generalization of the above mentioned Moon–Rand systems.