  Extreme solar activity fluctuations and the occurrence of solar grand minima
and maxima episodes, are well established, observed features of the solar
cycle. Nevertheless, such extreme activity fluctuations and the dynamics of the
solar cycle during Maunder minima-like episodes remain ill-understood. We
explore the origin of such extreme solar activity fluctuations and the role of
dual poloidal field sources, namely the Babcock-Leighton mechanism and the
mean-field alpha effect in the dynamics of the solar cycle. We mainly
concentrate on entry and recovery from grand minima episodes such as the
Maunder minimum and the dynamics of the solar cycle. We use a kinematic solar
dynamo model with a novel set-up in which stochastic perturbations force two
distinct poloidal field alpha effects. We explore different regimes of
operation of these poloidal sources with distinct operating thresholds, to
identify the importance of each. The perturbations are implemented
independently in both hemispheres which allows one to study the level of
hemispheric coupling and hemispheric asymmetry in the emergence of sunspots.
From the simulations performed we identify a few different ways in which the
dynamo can enter a grand minima episode. While fluctuations in any of the
$\alpha$ effects can trigger intermittency we find that the mean-field alpha
effect is crucial for the recovery of the solar cycle from a grand minima
episode which a Babcock-Leighton source alone, fails to achieve. Our
simulations also demonstrate other cycle dynamics. We conclude that stochastic
fluctuations in two interacting poloidal field sources working with distinct
operating thresholds is a viable candidate for triggering episodes of extreme
solar activity and that the mean-field alpha effect capable of working on weak,
sub-equipartition fields is critical to the recovery of the solar cycle
following an extended solar minimum.
