Singularities of periodic orbits near invariant curves

We show numerically, for standard-like maps, how the singularities (in the complex parameter) of the function which conjugates the map to a rotation of rational period behave when the period goes to an irrational number. Furthermore, we propose a numerical method to extrapolate the radius of convergence of the series parametrizing the solution of periodic orbits. The results are compared with analyses performed by Pade approximants, Greene's method, root criterion and the prediction by renormalization theory. (C) 2002 Elsevier Science B.V. All rights reserved.