  We study the late time evolution of positively curved FRW models with a
scalar field which arises in the conformal frame of the $R+\alpha R^{2}$
theory. The resulted three-dimensional dynamical system has two equilibrium
solutions corresponding to a de Sitter space and an ever expanding closed
universe. We analyze the structure of the first equilibrium with the methods of
the center manifold theory and, for the second equilibrium we apply the normal
form theory to obtain a simplified system, which we analyze with special phase
plane methods. It is shown that an initially expanding closed FRW spacetime
avoids recollapse.
