For autonomous C1-smooth integral equations with infinite delay, exponential attractivity with asymptotic phase of the local center manifolds of the equilibrium 0, together with a reduction principle, is proved by means of a dynamical systems approach based on the variation-of-constants formula in the phase space established in [Funkcial. Ekvac. 55(2012), 479–520]. As its application to one-parameter family of integral equations, it is also shown that saddle-node and pitchfork bifurcations occur when the equilibrium 0 (the zero solution) of the linearized equation changes its stability properties.
