This paper considers a nonlinear infinite-dimensional system ΣN obtained by the feedback interconnection of a well-posed linear system Σp with a globally Lipschitz (memoryless) nonlinear feedback operator N. First, under mild assumptions, we establish the global existence and uniqueness of a state trajectory and an output function for ΣN, for any initial state in its state space X and any input signal of class L2loc. Then we investigate the behavior of ΣN when it is driven by a nonlinear time-invariant exosystem with well defined dynamics forward and backward in time. Under the assumption that ΣP is exponentially stable, denoting the state space of the exosystem by W, we find that there exists a continuous map Π: W → X such that regardless of initial states limt→∞ ||Πw(t) - x(t)|| = 0, where w(t) is the state of the exosystem and x(t) is the state of ΣN. In particular, when w is T-periodic, then the state of the interconnection tends to a T-periodic limit cycle. The construction of Π can be viewed as an extension of the famous center manifold theorem, which lies at the basis of nonlinear regulator theory, to a class of infinite-dimensional systems.