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.
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