  We analyze the dynamics of the Bianchi I model in the presence of stiff
matter, an ultrarelativistic component and a small negative cosmological
constant. We quantize this model in the framework of the polymer quantum
mechanics, in order to introduce cut-off features in the minisuperspace
dynamics.
  We then apply to the polymer Wheeler-DeWitt equation, emerging from the Dirac
constraint, an adiabatic approximation a la Vilenkin, which treats the Universe
volume as a quasi-classical variable, becoming de facto the dynamical clock for
the pure quantum degrees of freedom, here identified in the Universe
anisotropies.
  The main issue of the present analysis consists of determining a cyclical
evolution for the Bianchi I model, oscillating between the Big-Bounce induced
by the cut-off physics and the turning point due to the small cosmological
constant. Furthermore, the mean value of the Universe anisotropy variables
remains finite during the whole evolution, including the phase across the
Big-Bounce. Such a feature, according to a suitable choice of the initial
conditions makes the present cosmological paradigm, a viable scenario for the
description of a possible primordial and late phases of the actual Universe.
