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A Physical Dissipative System with a Poincaré Homoclinic Figure-Eight

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Progress and Challenges in Dynamical Systems

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 54))

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Abstract

We consider 2D diffeomorphisms with a homoclinic figure-eight to a dissipative saddle under a periodic forcing. These systems are natural simplified models of phenomena with forcing and dissipation. As a physical example we study the dynamics of a parametrically driven dissipative pendulum with a magnetic kick forcing acting on it.

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Notes

  1. 1.

    We note that for large negative values of μ1,2 (depending on a + and a −) the origin becomes an stable focus and there is no longer a repellor inside the loops. In particular, the model does not verify the required assumptions (H1),…,(H6).

  2. 2.

    These values have been selected because the domains where the rich dynamics is expected are relatively large and the transient times to achieve the attractors not extremely long.

References

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Acknowledgements

This work has been supported by grants MTM2010-16425 (Spain) and 2009 SGR 67 (Catalonia). We thank J. Timoneda for the technical support on the computing facilities of the Dynamical Systems Group of the Universitat de Barcelona, largely used in this work.

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Authors and Affiliations

  1. Departament de Matemàtica Aplicada i Anàlisi, Universitat de Barcelona, Gran Via 585, 08007, Barcelona, Spain

    C. Simó & A. Vieiro

Authors
  1. C. Simó
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  2. A. Vieiro
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Corresponding author

Correspondence to C. Simó .

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Editors and Affiliations

  1. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Santiago Ibáñez

  2. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Jesús S. Pérez del Río

  3. Department of Mathematics, University of Oviedo, Oviedo, Spain

    Antonio Pumariño

  4. Department of Mathematics, University of Oviedo, Oviedo, Spain

    J. Ángel Rodríguez

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Simó, C., Vieiro, A. (2013). A Physical Dissipative System with a Poincaré Homoclinic Figure-Eight. In: Ibáñez, S., Pérez del Río, J., Pumariño, A., Rodríguez, J. (eds) Progress and Challenges in Dynamical Systems. Springer Proceedings in Mathematics & Statistics, vol 54. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-38830-9_24

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