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Piecewise Linear Bidimensional Maps as Models of Return Maps for 3D Diffeomorphisms

  • A. PumariñoEmail author
  • J. A. Rodríguez
  • J. C. Tatjer
  • E. Vigil
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)

Abstract

The goal of this paper is to study the dynamics of a simple family of piecewise linear maps in dimension two, that we call Expanding Baker Maps (EBM), which is a simplified model of a quadratic limit return map which appears in the study of certain homoclinic bifurcations of two-parameter families of three-dimensional dissipative diffeomorphisms. In spite of its simplicity the EBM capture some of the more relevant dynamics of the quadratic family, specially that related to the evolution of 2D strange attractors.

Keywords

Lyapunov Exponent Periodic Point Strange Attractor Critical Line Positive Lyapunov Exponent 
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.

Notes

Acknowledgements

This work has been supported by the MEC grants MTM2009-09723 and MTM2011-22956 and the CIRIT grant 2009 SGR 67.

References

  1. 1.
    Gonchenko, S.V., Gonchenko, V.S., Tatjer, J.C.: Bifurcations of three-dimensional diffeomorphisms with non-simple quadratic homoclinic tangencies and generalized Hénon maps. Regul. Chaotic Dyn. 12, 233–266 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Mora, L., Viana, M.: Abundance of strange attractors. Acta Math. 171, 1–71 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Palis, J., Takens, F.: Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge (1993)zbMATHGoogle Scholar
  4. 4.
    Palis, J., Viana M.: High dimension diffeomorphisms displaying infinitely many sinks. Ann. Math. 140, 91–136 (1994)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Pumariño, A., Tatjer, J.C.: Dynamics near homoclinic bifurcations of three-dimensional dissipative diffeomorphisms. Nonlinearity 19, 2833–2852 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Pumariño, A., Tatjer, J.C.: Attractors for return maps near homoclinic tangencies of three-dimensional dissipative diffeomorphisms. Discret. Contin. Dyn. Syst. Ser. B 8, 971–1006 (2007)CrossRefzbMATHGoogle Scholar
  7. 7.
    Tatjer, J.C.: Three-dimensional dissipative diffeomorphisms with homoclinic tangencies. Ergod. Theory Dyn. Syst. 21, 249–302 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Tatjer, J.C., Simó, C.: Basins of attraction near homoclinic tangencies. Ergod. Theory Dyn. Syst. 14, 351–390 (1994)CrossRefzbMATHGoogle Scholar
  9. 9.
    Tsujii, M.: Absolutely continuous invariant measures for expanding piecewise linear maps. Invent. Math. 143, 349–373 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Viana, M.: Strange attractors in higher dimensions. Bol. Soc. Bras. Mat. 24, 13–62 (1993)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • A. Pumariño
    • 1
    Email author
  • J. A. Rodríguez
    • 1
  • J. C. Tatjer
    • 2
  • E. Vigil
    • 1
  1. 1.Departamento de MatemáticasUniversidad de OviedoOviedoSpain
  2. 2.Departamento de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

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