Hamiltonian maps in the complex plane

Abstract

The standard map is a nonintegrable discrete time analog of the vertical pendulum. Detailed calculations are presented and illustrated graphically for the standard map at the golden mean frequency. The functional dependence of the coordinate q on the canonical angle variable θ is analtyically continued into the complex θ-plane, where natural boundaries are found at constant absolute values of Im θ. The boundaries represent the appearance of chaotic motion in the complex plane. When the domain of analyticity shrinks to zero, the KAM invariant curve is destroyed. Two independent numerical methods with Fourier analysis in the angle variable were used, one based on a variation-annihilation method and the other on a double expansion. The results were further checked by direct solution of the complex equations of motion. The numerically simpler, but intrinsically complex, semipendulum and semistandard map are also studied. We conjecture that natural boundaries appear in the analogous analytic continuation of the invariant tori or KAM surfaces of general nonintegrable systems with analytic Hamiltonians.