  The origin of cycle-to-cycle variations in solar activity is currently the
focus of much interest. It has recently been pointed out that large individual
active regions with atypical properties can have a significant impact on the
long term behaviour of solar activity. We investigate this possibility in more
detail using a recently developed 2$\times$2D dynamo model of the solar
magnetic cycle. We find that even a single "rogue" bipolar magnetic region
(BMR) in the simulations can have a major effect on the further development of
solar activity cycles, boosting or suppressing the amplitude of subsequent
cycles. In extreme cases an individual BMR can completely halt the dynamo,
triggering a grand minimum. Rogue BMRs also have the potential to induce
significant hemispheric asymmetries in the solar cycle. To study the effect of
rogue BMRs in a more systematic manner, a series of dynamo simulations were
conducted, in which a large test BMR was manually introduced in the model at
various phases of cycles of different amplitudes. BMRs emerging in the rising
phase of a cycle can modify the amplitude of the ongoing cycle while BMRs
emerging in later phases will only impact subsequent cycles. In this model, the
strongest impact on the subsequent cycle occurs when the rogue BMR emerges
around cycle maximum at low latitudes but the BMR does not need to be strictly
cross-equatorial. Active regions emerging as far as $20^{\circ}$ from the
equator can still have a significant impact. We demonstrate that the combined
effect of the magnetic flux, tilt angle and polarity separation of the BMR on
the dynamo is via their contribution to the dipole moment, $\delta
D_{\mathrm{BMR}}$. Our results indicate that prediction of the amplitude,
starting epoch and duration of a cycle requires an accurate accounting of a
broad range of active regions emerging in the previous cycle.
