Centre and isochronicity conditions for systems with homogeneous nonlinearities

We study the centre-focus problem for systems with homogeneous nonlinearities. In the centre case we study the characterization of the isochronous centres. More explicitly, we derive six necessary conditions to be a centre and six necessary conditions to be an isochronous centre. These conditions are expressed in complex notation and are suitable to be utilized in any computer algebra system. By using this approach we get the necessary and sufficient conditions to be a centre or an isochronous centre for a general system with homogeneous nonlinearities of degree two or three. This last result is already known but we give a slightly different proof