inflation have been scrutinized [@Vilenkin85; @Baibosunov90; @14047349; @14061096; @14100631; @14116010; @151209324].

Any regular model Universe must be non-singular for an infinitesimal perturbation of the metric $${\cal L}_{\text{reg}} = - \frac{M^{2}}{6} \left( \frac{R}{M^2} \right)^n \, . \label{Lregular}$$ The possible regular solutions of the gravitational equations in the presence of a nonlinear increment of the four curvature to the Lagrangian density of the gravitational field can be written as [@BGS71] $$S_{\text{BGS}} = \frac{\Mp^2}{16 \pi} \int \sqrt{-g} \, d^4 x \left( R + M^2 f_{\text{c}}\left[ \frac{R}{M^2} \right] \right) \, , \label{BGSfR}$$ where $f_{\text{c}}(R/M^2)$ is a certain dimensionless function of the scalar curvature and $M$ is the characteristic mass. This work is motivated to obtain the bouncing Universe at $\tau = a t = 0$.

As the minimal extension of the Starobinsky model, the one with the logarithmic correction term is given by [@Starobinsky79; @Shore80; @Vilenkin85; @Baibosunov90; @14047349] $$\begin{aligned}
 S &= \frac{\Mp^2}{2} \int \sqrt{-g} \, d^4 x \left( R + \tilde{\alpha} R^2 + \tilde{\beta} R^2 \ln R \right) \nonumber \\ 
&=  \frac{\Mp^2}{2} \int \sqrt{-g} \, d^4 x R \left(1 + \alpha \frac{R}{M^2} + \beta \frac{R}{M^2} \ln \frac{R}{M^2} \right) \, , \label{SR2lnR} \end{aligned}$$ where $\alpha = \tilde{\alpha} + 2 \tilde{\beta} \ln M$ and $\beta = 