Journal of Statistical Physics, Vol. 66, Nos. 5/6, 1992, p. 1613-1630.
We consider KAM invariant curves for generalizations of the standard map of the form (x', y')= (x + y', y + ef(x)), where f(x) is an odd trigonometric polynomial. We study numerically their analytic properties by a Pade approximant method applied to the function which conjugates the dynamics to a rotation teta -> teta + omega In the complex e plane, natural boundaries of different shapes are found. In the complex teta plane the analyticity region appears to be a strip bounded by a natural boundary, whose width tends linearly to 0 as e tends to the critical value.