The problem of local linearizability of the planar linear center perturbed by cubic non-linearities in all generalities on the system parameters (14 parameters) is far from being solved. Synchronization problem [18, 5] consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this paper contains a compact region around a singular point at the origin in which lies periodic orbits with the same period (independently from the chosen initial conditions). In this paper, starting from a 5-parameters non isochronous Chouikha cubic system [8], we identify all possible monomial perturbations of degree d ∈ {2, 3} insuring local linearizability of the perturbed system.The necessary conditions are obtained by the Normal Forms method. These conditions are real algebraic equations (multivariate polynomials) in the parameters of the studied ordinary differential system. The efficient algorithm FGb [13] for computing the Gröbner basis is used. For the family studied in this paper, an exhaustive list of possible parameters values insuring local linearizability is established. All the found cases are already known in the literature but the contexts is different since our object is the syn-chronisation not the classification. This paper can be seen as a direct continuation of several new works concerned with the hinting of cubic isochronous centers in particular [8, 12, 15, 10, 6, 2, 3], also it can be considered as an adaptation of a qualitative theory method to a synchronization problem. Mathematics Subject Classification (2010). Primary 34C15, 34C25, 34C37.