Scaling properties for the radius of convergence of Lindstedt series: Generalized standard maps

Abstract

For a class of symplectic two-dimensional maps which generalize the standard map by allowing more general nonlinear terms, the radius of convergence of the Lindstedt series describing the homotopically non-trivial invariant curves is proved to satisfy a scaling law as the complexified rotation number tends to a rational value non-tangentially to the real axis, thus generalizing previous results of the authors. The function conjugating the dynamics to rotations by ω possesses a limit which is explicitly computed and related to hyperelliptic functions in the case of nonlinear terms which are trigonometric polynomials. The case of the standard map is shown to be non-generic.