Elliptic islands on the elliptical stadium

Pages: 663 - 674, Volume 7, Issue 4, October 2001

doi:10.3934/dcds.2001.7.663       Abstract        Full Text (1349.2K)       Related Articles

Sylvie Oliffson Kamphorst - Departamento de Matemática, ICEx, UFMG, C.P. 702, 30161–970, Belo Horizonte, Brazil (email)
Sônia Pinto de Carvalho - Departamento de Matemática, ICEx, UFMG, C.P. 702, 30161–970, Belo Horizonte, Brazil (email)

Abstract: The elliptical stadium is a plane region bounded by a curve constructed by joining two half-ellipses with axes $a > 1$ and $b = 1$, by two parallel segments of equal length $2h$. The corresponding billiard problem defines a two-parameter family of discrete dynamical systems through the maps $T_{a,h}$.
We investigate the existence of elliptic islands for a special family of periodic orbits of $T_{a,h}$. The hyperbolic character of those orbits were studied in [2] for $1 < a < \sqrt 2$ and here we look for the elliptical character for every $a > 1$.
We prove that, for $a < \sqrt 2$, the lower bound for chaos $h = H(a)$ found in [2] is the upper bound of ellipticity for this special family. For $a > \sqrt 2$ we prove that there is no upper bound on h for the existence of elliptic islands.

Keywords:  Billiards, elliptic islands.
Mathematics Subject Classification:  37E40, 70K42.

Received: June 2000;      Revised: February 2001;      Published: July 2001.