Amplitude Independent Frequency Synchroniser via Normal Forms

The problem of local linearizability of the planar linear center nonlinearly perturbed is far from being solved even for low degree nonlinearity (n ≥ 3). Synchronization problem [1, 2] consists in bringing appropriate modifications on a given system to obtain a desired dynamic. The desired phase portrait along this contribution contains a compact region around a singular point at the origin in which lie periodic orbits with the same period (independently from the chosen initial conditions). We aim to present the methodology (involving polar normal forms and Gröbner Basis) for tackling this problem using a nonlinear monomial state feedback. As an example, we overcome the challenge for a cubic system. This contribution can be seen as a direct continuation of several new works concerned with the hinting of linearizability conditions in particular [7, 5, 3, 4], it can be also considered as an adaptation of a qualitative theory method to a synchronization problem.