  We explore a reduced Babcock-Leighton (BL) dynamo model based on delay
differential equations using numerical bifurcation analysis. This model reveals
hysteresis, seen in the recent mean-field dynamo model and the direct numerical
simulations of turbulent dynamos. The BL model with 'magnetic noise' as an
additional weak-source of the poloidal field recovers the solar cycle every
time from grand minima, which BL source alone cannot do. The noise-incorporated
model exhibits a bimodal distribution of toroidal field energy confirming two
modes of solar activity. It also shows intermittency and reproduces phase space
collapse, an experimental signature of the Maunder Minimum. The occurrence
statistics of grand minima in our model agree reasonably well with the observed
statistics in the reconstructed sunspot number. Finally, we demonstrate that
the level of magnetic noise controls the duration of grand minima and even has
a handle over its waiting period, suggesting a triggering effect of grand
minima by the noise and thus shutting down the global dynamo. Therefore, we
conclude that the 'magnetic noise' due to small-scale turbulent dynamo action
(or other sources) plays a vital role even in Babcock-Leighton dynamo models.
