We consider cyclic nearest neighbor systems of differential delay equations, in which the coupling between neighbors possesses a monotonicity property. Using a discrete (integer-valued) Lyapunov function, we prove that the Poincaré-Bendixson theorem holds for such systems. We also obtain results on piecewise monotonicity and stability of periodic solutions of such systems.
Bibliographical noteFunding Information:
J. Mallet-Paret was supported in part by National Science Foundation Grant DMS-9310328, by Office of Naval Research Contract N00014-92-J-1481, and by Army Research Office Contract DAAH04-93-G-0198. G. R. Sell was supported in part by the Army Research Office and by the National Science Fondation. Both authors acknowledge the support of the Army High Performance Computing Research Center, and the Institute for Mathematics and its Applications, at the University of Minnesota; the Lefschetz Center for Dynamical Systems, at Brown University; and the Center for Dynamical Systems and Nonlinear Studies, at the Georgia Institute of Technology.