A new deterministic model for chaotic reversals

  • C. GissingerEmail author
Regular Article


We present a new chaotic system of three coupled ordinary differential equations, limited to quadratic nonlinear terms. A wide variety of dynamical regimes are reported. For some parameters, chaotic reversals of the amplitudes are produced by crisis-induced intermittency, following a mechanism different from what is generally observed in similar deterministic models. Despite its simplicity, this system therefore generates a rich dynamics, able to model more complex physical systems. In particular, a comparison with reversals of the magnetic field of the Earth shows a surprisingly good agreement, and highlights the relevance of deterministic chaos to describe geomagnetic field dynamics.


Statistical and Nonlinear Physics 


  1. 1.
    S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, Berlin, 1990)Google Scholar
  2. 2.
    E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, Cambridge, UK, 1993)Google Scholar
  3. 3.
    J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences (Springer, New York, 1982)Google Scholar
  4. 4.
    E.N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20, 130 (1963)ADSGoogle Scholar
  5. 5.
    O.E. Rossler, Phys. Rev. Lett. A 57, 397 (1976)ADSGoogle Scholar
  6. 6.
    T. Rikitake, Math. Proc. Camb. Philos. Soc. 54, 89 (1958)MathSciNetADSzbMATHCrossRefGoogle Scholar
  7. 7.
    J.C. Sprott, Phys. Rev. E 50, 647 (1994)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    C. Gissinger, E. Dormy, S. Fauve, Europhys. Lett. 90, 49001 (2010)ADSCrossRefGoogle Scholar
  9. 9.
    P. Nozière, Phys. Earth Planet. Inter. 17, 55 (1978)ADSCrossRefGoogle Scholar
  10. 10.
    D. Hughes, M.R.E. Proctor, Nonlinearity 3, 127 (1990)MathSciNetADSzbMATHCrossRefGoogle Scholar
  11. 11.
    P. Smith, Explaining Chaos (Cambridge University Press, Cambridge, UK, 1998)Google Scholar
  12. 12.
    C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. 48, 1507 (1982)MathSciNetADSCrossRefGoogle Scholar
  13. 13.
    C. Grebogi, E. Ott, F. Romeira, J.A. Yorke, Phys. Rev. A 36, 5365 (1987)MathSciNetADSCrossRefGoogle Scholar
  14. 14.
    I. Melbourne, M.R.E. Proctor, A.M. Rucklidge, Dynamo and Dynamics, a Mathematical Challenge, edited by P. Chossat, D. Armbruster, I. Oprea (Kluwer, Dordrecht, 2001), pp. 363–370Google Scholar
  15. 15.
    P. Hoyng, M.A.J.H. Ossendrijver, D. Schmidt, Geophys. Astrophys. Fluid Dyn. 94, 263 (2001)ADSCrossRefGoogle Scholar
  16. 16.
    F. Petrelis, S. Fauve, E. Dormy, J.P. Valet, Phys. Rev. Lett. 102, 144503 (2009)ADSCrossRefGoogle Scholar
  17. 17.
    J.P. Valet, L. Meynadier, Y. Guyodo, Nature 435, 802 (2005)ADSCrossRefGoogle Scholar
  18. 18.
    P.L. McFadden et al., J. Geophys. Res. 96, 3923 (1991)ADSCrossRefGoogle Scholar
  19. 19.
    G. Glatzmaier, P. Roberts, Phys. Earth Planet. Inter. 91, 63 (1995)CrossRefGoogle Scholar
  20. 20.
    F. Petrelis, S. Fauve, J. Phys.: Condens. Matter 20, 494203 (2008)CrossRefGoogle Scholar
  21. 21.
    R. Monchaux et al., Phys. Rev. Lett. 98, 044502 (2007)ADSCrossRefGoogle Scholar
  22. 22.
    M. Berhanu et al., Europhys. Lett. 77, 59001 (2007)ADSCrossRefGoogle Scholar
  23. 23.
    C. Gissinger, Ph.D. thesis, Pierre and Marie Curie University, 2010Google Scholar
  24. 24.
    M. Kono, Geoph. Res. Lett. 14, 21 (1987)ADSCrossRefGoogle Scholar
  25. 25.
    V. Carbone et al., Phys. Rev. Lett. 96, 128501 (2006)ADSCrossRefGoogle Scholar
  26. 26.
    S.C. Cande, D.V. Kent, J. Geophys. Res. 100, 6093 (1995)ADSCrossRefGoogle Scholar
  27. 27.
    V. Courtillot, P. Olson, Earth Planet. Sci. Lett. 260, 495 (2007)ADSCrossRefGoogle Scholar
  28. 28.
    G. Glatzmaier et al., Nature 401, 885 (1999)ADSCrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of Astrophysical Sciences/Princeton Plasma Physics LabPrinceton UniversityPrincetonUSA

Personalised recommendations