Chaos Removal in the R+qR² gravity: the Mixmaster model

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).