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Bifurcations and Symmetry-Breaking in Simple Models of Nonlinear Dynamos

Published online by Cambridge University Press:  19 July 2016

N.O. Weiss
N.O. Weiss*
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge Cambridge CB3 9EW, U.K.

Abstract

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Low-order models of nonlinear dynamos can be used to investigate generic properties of more realistic mean field dynamos. Reducing the partial differential equations to a set of ordinary differential equations makes it possible to explore the bifurcation structure in considerable detail and to compute unstable solutions as well as ones that are stable. Complicated time-dependent behaviour is typically associated with a homoclinic or heteroclinic bifurcation. Destruction of periodic orbits at saddles or saddle-foci gives rise to Lorenz-like or Shil'nikov-like chaotic oscillations, while destruction of a quasiperiodic orbit leads to aperiodically modulated cycles. Changes in spatial symmetry can also be investigated. The interaction between solutions (steady or periodic) with dipole and quadrupole symmetry gives rise to a complicated bifurcation structure, with several recognizably different mixed-mode solutions; similar behaviour has also been found in spherical dynamo models. These results have implications for the expected behaviour of stellar dynamos.

Type
6. General Aspects of Dynamo Theory
Information
Symposium - International Astronomical Union , Volume 157: The Cosmic Dynamo , 1993 , pp. 219 - 229
Copyright
Copyright © Kluwer 1993 

References

Allan, D.W.: 1958, Nature 182, 469 Google Scholar
Allan, D.W.: 1962, Math. Proc. Cam. Phil. Soc. 58, 671 CrossRefGoogle Scholar
Brandenburg, A., Krause, F., Meinel, R., Moss, D. and Tuominen, I.: 1989, Astron. Astrophys. 213, 411 Google Scholar
Bullard, E.C.: 1978, in Jorna, S., ed(s)., Topics in nonlinear dynamics , American Institute of Physics: New York, 373 Google Scholar
Cartwright, M.L. and Littlewood, J.E.: 1945, J. Lond. Math. Soc. 20, 180 CrossRefGoogle Scholar
Cook, A.E. and Roberts, P.H.: 1970, Math. Proc. Cam. Phil. Soc. 68, 547 Google Scholar
Feudel, U., Jansen, W. and Kurths, J.: 1992, J. Bifurcation and Chaos , in press Google Scholar
Gubbins, D. and Zhang, K.: 1992, Phys. Earth. Planet. Int. , in press Google Scholar
Guckenheimer, J.: 1981, in Swinney, H.L. and Gollub, J.P., ed(s)., Hydrodynamic instabilities and the transition to turbulence , Springer: Berlin, 215 Google Scholar
Guckenheimer, J. and Holmes, P.: 1983, Nonlinear oscillations, dynamical systems and bifurcations of vector fields , Springer: New York Google Scholar
Jennings, R.L.: 1991, Geophys. Astrophys. Fluid. Dyn 57, 147 Google Scholar
Jennings, R.L. and Weiss, N.O.: 1991, Mon. Not. Roy. Astron. Soc. 252, 249 Google Scholar
Jones, C.A., Weiss, N.O. and Cattaneo, F.: 1985, Physica 14D, 161 Google Scholar
Kirk, V.: 1991, Phys. Lett. A 154, 243 Google Scholar
Kirk, V.: 1993, Nonlinearity , submitted Google Scholar
Knobloch, E.: 1981, Phys. Lett. A 82, 439 Google Scholar
Knobloch, E.: 1993, in Proctor, M.R.E. and Gilbert, A. D., ed(s)., Theory of solar and planetary dynamos: introductory lectures , Cambridge University Press: Cambridge,Google Scholar
Knobloch, E., Proctor, M.R.E. and Weiss, N.O.: 1992, J. Fluid Mech. 239, 273 Google Scholar
Krause, F. and Roberts, P.H.: 1981, Adv. Space Res. 1, 231 Google Scholar
Langford, W.F.: 1983, in Barenblatt, G. I., Iooss, G. and Joseph, D. D., ed(s)., Nonlinear dynamics and turbulence , Pitman: London, 215 Google Scholar
Lorenz, E.N.: 1963, J. Atmos. Sci. 20, 130 2.0.CO;2>CrossRefGoogle Scholar
Malkus, W.V.R.: 1972, EOS, Trans. Amer. Geophys. Union 53, 617 Google Scholar
Moss, D., Tuominen, I. and Brandenburg, A.: 1990, Astron. Astrophys. 228, 284 Google Scholar
Noyes, R.W., Weiss, N.O. and Vaughan, A.H.: 1984, Astrophys. J. 287, 769 Google Scholar
Rädler, K.-H., Wiedemann, E., Brandenburg, A., Meinel, R. and Tuominen, I.: 1990, Astron. Astrophys. 239, 413 Google Scholar
Robbins, K.A.: 1977, Math. Proc. Cam. Phil. Soc. 82, 309 Google Scholar
Schmalz, S. and Stix, M.: 1991, Astron. Astrophys. 245, 654 Google Scholar
Schmitt, D. and Schüssler, M.: 1989, Astron. Astrophys. 223, 343 Google Scholar
Shil'nikov, L.P.: 1965, Sov. Math. Dokl. 6, 163 Google Scholar
Smale, S.: 1963, in Cairns, S. S., ed(s)., Differential and combinatorial topology , Princeton University Press: Princeton, 63 Google Scholar
Smale, S.: 1967, Bull. Am. Math. Soc. 73, 747 CrossRefGoogle Scholar
Sparrow, C.: 1982, The Lorenz equations: bifurcations, chaos and strange attractors , Springer: New York CrossRefGoogle Scholar
Stix, M.: 1972, Astron. Astrophys. 20, 9 Google Scholar
Weiss, N.O., Cattaneo, F. and Jones, C.A.: 1984, Geophys. Astrophys. Fluid. Dyn. 30, 305 Google Scholar
Wiggins, S.: 1988, Global bifurcations and chaos , Springer: New York Google Scholar
Zel'dovich, Ya.B., Ruzmaikin, A.A. and Sokoloff, D.D.: 1983, Magnetic fields in astrophysics , Gordon and Breach: London Google Scholar