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Theory of gravity with a quadratic contribution of scalar curvature isinvestigated in terms of dynamical system approach. The simplestFriedmann-Robertson-Walker metric is used to formulate dynamics in Jordan frameas well as in conformally transformed Einstein frame. We show that in bothframes there are stable de Sitter states for which the Hubble functionexpansion naturally gives terms corresponding to non-substantial dark matter.Using invariant centre manifold we show that in the Einstein frame there is azero measure set of initial conditions leading from unstable to stable deSitter state. Additionally, the initial de Sitter state is plunged with aparallelly propagated singularity. We show that the Jordan frame and theEinstein frame formulation of the theory are physically nonequivalent.

On the structural stability of a simple cosmological model in $R+\alpha R^{2}$ theory of gravity