  We study the asymptotic dynamics of the Mixmaster Universe, near the
cosmological singularity, considering $f(R)$ gravity up to a quadratic
corrections in the Ricci scalar $R$. The analysis is performed in the
scalar-tensor framework and adopting Misner-Chitr\'e-like variables to describe
the Mixmaster Universe, whose dynamics resembles asymptotically a billiard-ball
in a given domain of the half-Poincar\'e space. The form of the potential well
depends on the spatial curvature of the model and on the particular form of the
self-interacting scalar field potential. We demonstrate that the potential
walls determine an open domain in the configuration region, allowing the
point-Universe to reach the absolute of the considered Lobachevsky space. In
other words, we outline the existence of a stable final Kasner regime in the
Mixmaster evolution, implying the chaos removal near the cosmological
singularity. The relevance of the present issue relies both on the general
nature of the considered dynamics, allowing its direct extension to the BKL
conjecture too, as well as the possibility to regard the considered modified
theory of gravity as the first correction to the Einstein-Hilbert action as a
Taylor expansion of a generic function $f(R)$ (as soon as a cut-off on the
space-time curvature takes place).
