Starobinsky cosmological model in Palatini formalism
  We classify singularities in FRW cosmologies, which dynamics can be reduced
to the dynamical system of the Newtonian type. This classification is performed
in terms of geometry of a potential function if it has poles. At the sewn
singularity, which is of a finite scale factor type, the singularity in the
past meets the singularity in the future. We show, that such singularities
appear in the Starobinsky model in $f(\hat{R})=\hat{R}+\gamma \hat{R}^2$ in the
Palatini formalism, when dynamics is determined by the corresponding piece-wise
smooth dynamical system. As an effect we obtain a degenerated singularity.
Analytical calculations are given for the cosmological model with matter and
the cosmological constant. The dynamics of model is also studied using
dynamical system methods. From the phase portraits we find generic evolutionary
scenarios of the evolution of the Universe. For this model, the best fit value
of $\Omega_\gamma=3\gamma H_0^2$ is equal $9.70\times 10^{-11}$. We consider
model in both Jordan and Einstein frames. We show that after transition to the
Einstein frame we obtain both form of the potential of the scalar field and the
decaying Lambda term.
