Do sewn singularities falsify the Palatini cosmology?
  We investigate further (cf. arXiv:1512.01199, JCAP01 (2016) 040) Starobinsky
cosmological model $R+\gamma R^2$ in the Palatini formalism with Chaplygin gas
and baryonic matter as a source. For this aim we use dynamical system theory.
The dynamics is reduced to the 2D sewn dynamical system of a Newtonian type (a
piecewise-smooth dynamical system). We classify all evolutional paths in the
model as well as trajectories in the phase space. We demonstrate the presence
of a degenerate freeze singularity (glued freeze type singularities) for the
positive $\gamma$. In this case it is a generic feature of early evolution of
the universe. We point out that a degenerate type III of singularity can be
considered as an endogenous model of inflation between the matter dominating
epoch and the dark energy phase. We also investigate cosmological models with
negative $\gamma$. It is demonstrated that $\gamma$ equal zero is a bifurcation
parameter and dynamics qualitatively changes in comparison to positive
$\gamma$. Instead of the big bang the sudden bounce singularity of a finite
scale factor appears and there is a generic class of bouncing solutions sewn
along the line $a=a_{\text{sing}}$. And we argue that the presence of sudden
singularities in an evolutional scenario of the Universe falsifies the negative
$\gamma$ in the Palatini cosmology. Only very small values of $\Omega_{\gamma}$
parameter are admissible if we requires that agreements physics with the
$\Lambda$CDM model. From the statistical analysis of astronomical observations,
we deduce that the case of negative values of $\Omega_\gamma$ can be rejected
even if it may fit better to the data.
