  We examine kinematic dynamo action driven by an axisymmetric large scale flow
that is superimposed with an azimuthally propagating non-axisymmetric
perturbation with a frequency $\omega$. Although we apply a rather simple large
scale velocity field, our simulations exhibit a complex behavior with
oscillating and azimuthally drifting eigenmodes as well as stationary regimes.
Within these non-oscillating regimes we find parametric resonances
characterized by a considerable enhancement of dynamo action and by a locking
of the phase of the magnetic field to the pattern of the perturbation. We find
an approximate fulfillment of the relationship between the resonant frequency
$\omega_{\rm{res}}$ of the disturbed system and the eigenfrequency $\omega_0$
of the undisturbed system given by $\omega_{\rm{res}}=2\omega_0$ which is known
from paradigmatic rotating mechanical systems and our prior study [Giesecke et
al., Phys. Rev. E, 86, 066303 (2012)]. We find further -- broader -- regimes
with weaker enhancement of the growth rates but without phase locking. However,
this amplification regime arises only in case of a basic (i.e. unperturbed)
state consisting of several different eigenmodes with rather close growth
rates. Qualitatively, these observations can be explained in terms of a simple
low dimensional model for the magnetic field amplitude that is derived using
Floquet theory. The observed phenomena may be of fundamental importance in
planetary dynamo models with the base flow being disturbed by periodic external
forces like precession or tides and for the realization of dynamo action under
laboratory conditions where imposed perturbations with the appropriate
frequency might facilitate the occurrence of dynamo action.
