In this paper, we deal with a classical object, namely, a nonhyperbolic limit
cycle in a system of smooth autonomous ordinary differential equations. While
the existence of a center manifold near such a cycle was assumed in several
studies on cycle bifurcations based on periodic normal forms, no proofs were
available in the literature until recently. The main goal of this paper is to
give an elementary proof of the existence of a periodic smooth locally
invariant center manifold near a nonhyperbolic cycle in finite-dimensional
ordinary differential equations by using the Lyapunov-Perron method. In
addition, we provide several explicit examples of analytic vector fields
admitting (non)-unique, (non)-Cβ-smooth and (non)-analytic periodic
center manifolds.Comment: 35 pages, 4 figure