  Context. Solar activity cycles vary in amplitude and duration. The variations
can be at least partly explained by fluctuations in dynamo parameters. Aims. We
want to restrict uncertainty in fluctuating dynamo parameters and find out
which properties of the fluctuations control the amplitudes of the magnetic
field and energy in variable dynamo cycles. Methods. A flux-transport model for
the solar dynamo with fluctuations of the Babcock-Leighton type $\alpha$-effect
was applied to generate statistics of magnetic cycles for our purposes. The
statistics were compared with data on solar cycle periods to restrict the
correlation time of dynamo fluctuations. Results. A characteristic time of
fluctuations in the $\alpha$-effect is estimated to be close to the solar
rotation period. The fluctuations produce asymmetry between the times of rise
and descent of dynamo cycles, the rise time being on average shorter. The
affect of the fluctuations on cycle amplitudes depends on the phase of the
cycle in which the fluctuations occur. Negative fluctuations (decrease in
$\alpha$) in the rise phase delay decay of poloidal field and increase the
cycle amplitude in toroidal field and magnetic energy. Negative fluctuation in
the decline phase reduces the polar field at the end of a cycle and the
amplitude of the next cycle. The low amplitude of the 24th solar cycle compared
to the preceding 23rd cycle can be explained by this effect. Positive
fluctuations in the descent phase enhance the magnetic energy of the next cycle
by increasing the seed poloidal field for the next cycle. The statistics of the
computed energies of the cycles suggest that superflares of $\ge 10^{34}$ erg
are not possible on the Sun.
