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Communications in Mathematical Physics

, Volume 174, Issue 3, pp 661–679 | Cite as

Chaotic properties of the elliptical stadium

  • Roberto Markarian
  • Sylvie Oliffson Kamphorst
  • Sônia Pinto de Carvalho
Article

Abstract

The elliptical stadium is a curve constructed by joining two half-ellipses, with half axesa>1 andb=1, by two straight segments of equal length 2h.

Donnay [6] has shown that if 1 <a <\(\sqrt 2\) and ifh is big enough, then the corresponding billiard map has a positive Lyapunov exponent almost everywhere; moreover,h→∞ asa\(\sqrt 2\)

In this work we prove that if\(1< a< \sqrt {4 - 2\sqrt 2 }\), then\(h > 2a^2 \sqrt {a^2 - 1}\) assures the positiveness of a Lyapunov exponent. And we conclude that, for these values ofa andh, the elliptical stadium billiard mapping is ergodic and has theK-property.

Keywords

Neural Network Statistical Physic Assure Complex System Nonlinear Dynamics 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Roberto Markarian
    • 1
  • Sylvie Oliffson Kamphorst
    • 2
  • Sônia Pinto de Carvalho
    • 2
  1. 1.Instituto de Matemática y Estadística “Prof. Ing. Rafael LanguardiaFacultad de IngenieríaMontevideoUruguay
  2. 2.Departmento de MatemáticaICEx, UFMGBelo HorizonteBrasil

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