Do spherical alpha^2-dynamos oscillate?
  The question is answered whether kinematic alpha^2-shell-dynamos are able to produce a cyclic activity or not. The alpha-effect is allowed to be latitudinally inhomogeneous and/or anisotropic, but it is assumed as radially uniform in the turbulent shell.   For a symmetric alpha-tensor we only find oscillatory solutions if i) the alpha_zz vanishes or is of the opposite sign as alpha_{phi phi}, ii) the alpha-effect is strongly concentrated to the equatorial region and iii) the alpha-effect is concentrated to a rather thin outer shell. In the other cases almost always the nonaxisymmetric field mode S1 possesses the lowest dynamo number which slowly drifts along the azimuthal direction. Also uniform but anisotropic alpha-effect (alpha_zz = 0) leads to the nonaxisymmetric solutions as it is confirmed by the Karlsruhe dynamo experiment. However, one of the antisymmetric parts of the alpha-tensor basically plays the role of a differential rotation in the induction equation. Using for the radial profile of this effect the results of anumerical simulation for the alpha-tensor of the solar convection zone, one indeed finds the possibility of oscillating alpha^2-dynamos even without the existence of real nonuniform plasma rotation. 
G. R\"udiger, D. Elstner, M. Ossendrijver