Abstract
In their pioneering work BUSSE & HEIKES [2] looked at the classical Bénard problem in a rotating frame. They observed rolls with a certain axial direction which seem to be stable, ie they remain unchanged for long periods of time, but suddenly the behavior changes: new rolls appear which are rotated with respect to the original rolls by approximately 60 degrees. GUCKENHEIMER & HOLMES [15] looked at the Busse-Heikes problem from a theoretical point of view. They derived a three dimensional ODE exhibiting heteroclinic cycles. Thereafter many papers have dealt with various aspects of cycles: existence, stability, bifurcations and structural stability under certain settings: [1, 3, 4, 5, 7, 8, 16, 17, 18, 20, 21, 22, 25, 23, 24, 26, 27, 28, 29]. Due to the fact that solutions which pass near steady states remain there for a long time heteroclinic cycles serve as a model for metastable behavior. We see such a metastable behavior if we look at the polarity reversals of the magnetic field of the Earth, see for example GHIL & CHILDRESS [13]. Since the origin of the magnetic field and the mechanism of its reversals are unknown we try to look at it from a dynamical systems point of view and ask ourselves whether there are heteroclinic cycles in this problem. Here, we only look at the fluid mechanical part and not at full MHD-equations. In this paper we emphasize the role of rotation and the question what happens to heteroclinic cycles in the presence of rotations. Of course this approach dictates to look at the non rotating case first and treat the rotating case as a perturbation of the non rotating one. It is not clear whether such a approach is reasonable for studying the Earth’ field but on the other hand numerical computations indicate that the region of validity of the results which are presented here exceeds the marginal speeds of rotation allowed by the usual perturbation methods.
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Lauterbach, R. (2001). Heteroclinic Cycles and Fluid Motions in Rotating Spheres. In: Chossat, P., Ambruster, D., Oprea, I. (eds) Dynamo and Dynamics, a Mathematical Challenge. NATO Science Series, vol 26. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0788-7_41
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