Abstract
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.
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References
- 1.
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, Berlin, 1990)
- 2.
E. Ott, Chaos in Dynamical Systems (Cambridge Univ. Press, Cambridge, UK, 1993)
- 3.
J. Guckenheimer, P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, Applied Mathematical Sciences (Springer, New York, 1982)
- 4.
E.N. Lorenz, Deterministic non-periodic flow, J. Atmos. Sci. 20, 130 (1963)
- 5.
O.E. Rossler, Phys. Rev. Lett. A 57, 397 (1976)
- 6.
T. Rikitake, Math. Proc. Camb. Philos. Soc. 54, 89 (1958)
- 7.
J.C. Sprott, Phys. Rev. E 50, 647 (1994)
- 8.
C. Gissinger, E. Dormy, S. Fauve, Europhys. Lett. 90, 49001 (2010)
- 9.
P. Nozière, Phys. Earth Planet. Inter. 17, 55 (1978)
- 10.
D. Hughes, M.R.E. Proctor, Nonlinearity 3, 127 (1990)
- 11.
P. Smith, Explaining Chaos (Cambridge University Press, Cambridge, UK, 1998)
- 12.
C. Grebogi, E. Ott, J.A. Yorke, Phys. Rev. Lett. 48, 1507 (1982)
- 13.
C. Grebogi, E. Ott, F. Romeira, J.A. Yorke, Phys. Rev. A 36, 5365 (1987)
- 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. 363–370
- 15.
P. Hoyng, M.A.J.H. Ossendrijver, D. Schmidt, Geophys. Astrophys. Fluid Dyn. 94, 263 (2001)
- 16.
F. Petrelis, S. Fauve, E. Dormy, J.P. Valet, Phys. Rev. Lett. 102, 144503 (2009)
- 17.
J.P. Valet, L. Meynadier, Y. Guyodo, Nature 435, 802 (2005)
- 18.
P.L. McFadden et al., J. Geophys. Res. 96, 3923 (1991)
- 19.
G. Glatzmaier, P. Roberts, Phys. Earth Planet. Inter. 91, 63 (1995)
- 20.
F. Petrelis, S. Fauve, J. Phys.: Condens. Matter 20, 494203 (2008)
- 21.
R. Monchaux et al., Phys. Rev. Lett. 98, 044502 (2007)
- 22.
M. Berhanu et al., Europhys. Lett. 77, 59001 (2007)
- 23.
C. Gissinger, Ph.D. thesis, Pierre and Marie Curie University, 2010
- 24.
M. Kono, Geoph. Res. Lett. 14, 21 (1987)
- 25.
V. Carbone et al., Phys. Rev. Lett. 96, 128501 (2006)
- 26.
S.C. Cande, D.V. Kent, J. Geophys. Res. 100, 6093 (1995)
- 27.
V. Courtillot, P. Olson, Earth Planet. Sci. Lett. 260, 495 (2007)
- 28.
G. Glatzmaier et al., Nature 401, 885 (1999)
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Gissinger, C. A new deterministic model for chaotic reversals. Eur. Phys. J. B 85, 137 (2012). https://doi.org/10.1140/epjb/e2012-20799-5
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Keywords
- Statistical and Nonlinear Physics