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October  2001, 7(4): 663-674. doi: 10.3934/dcds.2001.7.663

Elliptic islands on the elliptical stadium

Sylvie Oliffson Kamphorst 1, and  Sônia Pinto de Carvalho 1,

1. 

Departamento de Matemática, ICEx, UFMG, C.P. 702, 30161–970, Belo Horizonte, Brazil, Brazil

Received  June 2000 Revised  February 2001 Published  July 2001

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: elliptic islands., Billiards.
Mathematics Subject Classification: 37E40, 70K4.
Citation: Sylvie Oliffson Kamphorst, Sônia Pinto de Carvalho. Elliptic islands on the elliptical stadium. Discrete and Continuous Dynamical Systems, 2001, 7 (4) : 663-674. doi: 10.3934/dcds.2001.7.663
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