A Simple Family of Exceptional Maps with Chaotic Behavior

• Martínez, Regina ^[1] ; Simó Carles ^[2]
  1. [1] Universitat Autònoma de Barcelona

     Universitat Autònoma de Barcelona

     Barcelona, España

     
  2. [2] Universitat de Barcelona

     Universitat de Barcelona

     Barcelona, España

     
• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 19, Nº 1, 2020
• Idioma: inglés
• DOI: 10.1007/s12346-020-00361-w
• Enlaces
  • Texto completo
• Resumen

  • A simple family of maps in T2 is considered in this note. It displays chaos in the sense that the dynamics has sensitive dependence to initial conditions and topological transitivity. Furthermore the set of points displaying chaotic behavior has full Lebesgue measure in T2. However the maps have neither homoclinic nor heteroclinic orbits and have a single fixed point which is parabolic, with an unstable branch and a stable one. The role of returning infinitely many times near the fixed point is taken by quasi-periodicity. The maximal Lyapunov exponent is zero. This family was presented as a one-page example in Garrido and Simó (Some ideas about strange attractors. Dynamical systems and chaos (Sitges/Barcelona, 1982). Lecture notes in physics, Springer, Berlin, 1983) (section 2.8). Later we present generalizations and variants.

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