Advertisement

A Physical Dissipative System with a Poincaré Homoclinic Figure-Eight

  • C. SimóEmail author
  • A. Vieiro
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 54)

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.

Keywords

Bifurcation Diagram Invariant Manifold Invariant Curve Bifurcation Problem Invariant Curf 
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 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.

References

  1. 1.
    Afraimovich, V.S., Shilnikov, L.P.: The annulus principle and problems on interaction of two self-oscillation systems. Prikladnaja Matematica i Mekhanika 41, 618–627 (1977)MathSciNetGoogle Scholar
  2. 2.
    Broer, H., Simó, C., Tatjer J.C.: Towards global models near homoclinic tangencies of dissipative diffeomorphisms. Nonlinearity 11, 667–770 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Chirikov, B.V.: A universal instability of many-dimensional oscillator system. Phys. Rep. 52, 264–379 (1979)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Gonchenko, S.V., Simó, C., Vieiro, A.: A global study of 2D dissipative diffeomorphisms with a Poincaré homoclinic figure-eight. Nonlinearity 26(3), 621–679 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Kuznetsov, Y.A.: Elements of applied bifurcation theory. In: Applied Mathematical Sciences, vol. 112, 3rd edn. Springer, New York (2004)Google Scholar
  6. 6.
    Turaev, D.V.: On a case of bifurcations of a contour composed by two homoclinic curves of a saddle. In: Methods of the Qualitative Theory of Differential Equations, pp. 162–175. Gorki State University Pub. (1984) (in Russian)Google Scholar
  7. 7.
    Turaev, D.V., Shilnikov, L.P.: On bifurcations of a homoclinic figure-eight for a saddle with a negative saddle value. Sov. Math. Dokl. 44(2), 422–426 (1987)MathSciNetGoogle Scholar
  8. 8.
    Zaslavsky, G.M., Filonenko, N.N.: Stochastic instability of trapped particles and conditions of applicability of the quasi-linear approximation. Sov. Phys. JETP 27, 851–857 (1968)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Departament de Matemàtica Aplicada i AnàlisiUniversitat de BarcelonaBarcelonaSpain

Personalised recommendations