Periodic solutions for an N-dimensional cyclic feedback system with delay

We study models for N cyclically coupled variables (e.g., neuron activities) with overall negative delayed feedback, and without symmetry or monotonicity properties. Our aim is to extract the common parts of similar approaches that are known in dimensions one, two and three so far, to exhibit how these parts work for general dimension N, and to show how this framework includes previous as well as new results. We provide a fixed point theorem and a related theorem on periodic orbits for semiflows on Banach spaces, which then yield periodic solutions of cyclic delayed negative feedback systems for general N. We also give criteria for the global asymptotic stability in the same systems, which are derived by relating the systems to interval maps.