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

Scalar field cosmology -- toward description of dynamic complexity of cosmological evolution