We define a discrete (integer-valued) Lyapunov function V for cyclic nearest neighbor systems of differential delay equations possessing a feedback condition. This extends analogous definitions for cyclic systems of ODE's, and for scalar differential delay equations. We relate the values of V to the real parts of the Floquet multipliers for such linear periodic systems, and thereby prove all Floquet subspaces are at most two-dimensional.
Bibliographical noteFunding Information:
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 Foundation. 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.