  We study the dynamical evolution of cosmological models with the
Robertson-Walker symmetry with a scalar field non-minimally coupled to gravity
and barotropic matter. For this aim we use dynamical system methods. We have
found a type of evolutional path which links between all important events
during the evolution, the cosmological singularity of finite time, inflation,
radiation and matter dominating epoch and the accelerated phase expansion of
the universe. We point out importance of finding the new generic solution
called a twister solution for a deeper description of the evolution of the
Universe. We demonstrate that including the non-minimal coupling leads to a
new, richer evolutional cosmological scenario in comparison to the case of
minimal coupling.
