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Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos

Kirillov, O. N. and Günther, U. and Stefani, F. (2009):
Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos.
In: Physical Review E 79, [Online-Edition: http://link.aps.org/doi/10.1103/PhysRevE.79.016205],
[Article]

Item Type: Article
Erschienen: 2009
Creators: Kirillov, O. N. and Günther, U. and Stefani, F.
Title: Determining role of Krein signature for three-dimensional Arnold tongues of oscillatory dynamos
Language: English
Journal or Publication Title: Physical Review E 79
Divisions: 16 Department of Mechanical Engineering
16 Department of Mechanical Engineering > Dynamics and Vibrations
Date Deposited: 19 Sep 2012 14:21
Official URL: http://link.aps.org/doi/10.1103/PhysRevE.79.016205
Identification Number: doi:10.1103/PhysRevE.79.016205
Alternative Abstract:
Alternative abstract Language
Using a homotopic family of boundary eigenvalue problems for the mean-field α2 dynamo with helical turbulence parameter α(r)=α0+γΔα(r) and homotopy parameter β∊[0,1], we show that the underlying network of diabolical points for Dirichlet (idealized, β=0) boundary conditions substantially determines the choreography of eigenvalues and thus the character of the dynamo instability for Robin (physically realistic, β=1) boundary conditions. In the (α0,β,γ) space the Arnold tongues of oscillatory solutions at β=1 end up at the diabolical points for β=0. In the vicinity of the diabolical points the space orientation of the three-dimensional tongues, which are cones in first-order approximation, is determined by the Krein signature of the modes involved in the diabolical crossings at the apexes of the cones. The Krein space-induced geometry of the resonance zones explains the subtleties in finding α profiles leading to spectral exceptional points, which are important ingredients in recent theories of polarity reversals of the geomagnetic field.English
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