  We seek to understand the transition from nearly axisymmetric configurations
at solar rotation rates to nonaxisymmetric configurations for rapid rotation
using 3D numerical simulations of turbulent convection and considering rotation
rates between 1 and 30 times the solar value. We find a transition from axi- to
nonaxisymmetric solutions at around 1.8 times the solar rotation rate. This
transition coincides with a change in the rotation profile from antisolar- to
solar-like differential rotation with a faster equator and slow poles. In the
solar-like rotation regime, the field configuration consists of an axisymmetric
oscillatory field accompanied by an m=1 azimuthal mode (two active longitudes),
which also shows temporal variability. At slow (rapid) rotation, the
axisymmetric (nonaxisymmetric) mode dominates. The axisymmetric mode produces
latitudinal dynamo waves with polarity reversals, while the nonaxisymmetric
mode often exhibits a drift in the rotating reference frame and the strength of
the active longitudes changes cyclically over time between the different
hemispheres. Most of the obtained dynamo solutions exhibit cyclic variability
either caused by latitudinal or azimuthal dynamo waves. In an activity-period
diagram, the cycle lengths normalized by the rotation period form two different
populations as a function of rotation rate or magnetic activity level. The
slowly rotating axisymmetric population lies close to what is called the
inactive branch in observations, while the rapidly rotating models are close to
the superactive branch with a declining cycle to rotation frequency ratio with
increasing rotation rate. We can successfully reproduce the transition from
axi- to nonaxisymmetric dynamo solutions for high rotation rates, but
high-resolution simulations are required to limit the effect of rotational
quenching of convection at rotation rates above 20 times the solar value.
