Discretization and Morse { Smaledynamical Systems on Planar

In a previous paper 12], we have shown that locally, in the vicinity of hyperbolic equilibria of autonomous ordinary diierential equations, the time-h-map of the induced dynamical system is conjugate to the h-discretized system i.e. to the discrete dynamical system obtained via one-step discretization with stepsize h. The present paper is devoted to Morse-Smale dynamical systems deened on planar discs and having no periodic orbits. Using elementary extension techniques, we point out that local conjugacies about saddle points can be glued together: the time-h-map is globally conjugate to the h-discretized system. This is a discretization analogue of the famous Andronov-Pontryagin theorem 2], 18] on structural stability. For methods of order p, the conjugacy is O(h p)-near to the identity. The paper ends with some general remarks on similar problems.