FM 01-14

Author: Alberto Berretti, Guido Gentile

Title: Periodic and quasi-periodic orbits for the standard map
Abstract: We consider both periodic and quasi-periodic solutions for the standard map, and we study the corresponding conjugating functions, i.e. the functions conjugating the motions to trivial rotations. We compare the invariant curves with rotation numbers satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents We prove the following results: (1) for rotation numbers corresponding to the convergents we study the radius of convergence of the conjugating functions and we find lower bounds on them, which tend to a limit which is a lower bound on the corresponding quantity for the Bryuno rotation number; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number to which the convergents tend; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve.

Keywords: Standard map, perturbation theory, resonances, periodic orbits, invariant curves

Alberto Berretti
Dipartimento di Matematica
II Universita` di Roma (Tor Vergata)
Via della Ricerca Scientifica, 00133 Roma, Italy

Guido Gentile
Dipartimento di Matematica
Universita` di Roma Tre
Largo San Leonardo Murialdo 1, 00146 Roma, Italy