Different methods are proposed and tested for transforming a non linear differential system, and more particulary a hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincar\'e return map technique. We construct piecewise polynomial maps by coarse-graining the phase surface of section into parallelograms using values of the Poincar\'e maps at the vertices to define a polynomial approximation within each cell. The numerical experiments are in good agreement with both the real symplectic and Poincar\'e maps. The agreement is better when the number of vertices and the order of the polynomial fit increase. Computations of Lyapunov Characteristic Exponents give a measure of how well the fit approximate the different maps.