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 ω satisfying the Bryuno condition and the sequences of periodic orbits with rotation numbers given by their convergents ω _N = p _N /q _N . We prove the following results for N→ ∞: (1) for rotation numbers ω _N N 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 ω; (2) the periodic orbits consist of points which are more and more close to the invariant curve with rotation number ω; (3) such orbits lie on analytical curves which tend uniformly to the invariant curve.