Attractivity with asymptotic phase of local center manifolds and an application to one-parameter bifurcation for integral equations with infinite delay

For autonomous <a:math xmlns:a="http://www.w3.org/1998/Math/MathML"> <a:msup> <a:mi>C</a:mi> <a:mn>1</a:mn> </a:msup> </a:math>-smooth integral equations with infinite delay, exponential attractivity with asymptotic phase of the local center manifolds of the equilibrium <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"> <b:mn>0</b:mn> </b:math>, 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 <c:math xmlns:c="http://www.w3.org/1998/Math/MathML"> <c:mn>0</c:mn> </c:math> (the zero solution) of the linearized equation changes its stability properties.