  In this work we use a low order dynamo model and study under which conditions
can it reproduce solar grand minima. We begin by building the phase space of a
proxy for the toroidal component of the solar magnetic field and we develop a
model, derived from mean field dynamo theory, that gives the time evolution of
the toroidal field. This model is characterized by a non-linear oscillator
whose coefficients retain most of the physics behind dynamo theory. In the
derivation of the model we also include stochastic oscillations in the $\alpha$
effect. We found no evidences that stochastic fluctuations in a linear $\alpha$
effect can trigger grand minima episodes in this model. In contrast, the model
used points out that possible mechanism that can trigger grand minima should
involve the meridional circulation, magnetic diffusivity or field
intensification by buoyancy driven instabilities.
