Abstract. In this paper, we present a method for the analysis of ordinary differential equations. The main idea is to replace the original non-linear dynamical system by an approximate piecewise linear one. The latter gives an algebraic expression of the solutions. Our method differs also from classical methods (e.g. Runge-Kutta) by the fact that we use a discretization of the phase space instead of the time space. Theoretical results of convergence exist and are proved here. An efficient symbolic-numeric algorithm for the construction of the solutions of this piecewise linear system is given. As an application, we show how we can compute an accurate approximation of periodic orbits using symbolic differentiation of the Poincaré map.