Differential Systems with Feedback: Time Discretizations and Lyapunov Functions

In this paper we examine a class of Eulerian time discretizations for a monotone cyclic feedback system with a time delay; see Mallet-Paret and Sell (1996a, 1996b) for background information. We construct an integer-valued function V for the discrete-time problem. The Main Theorem shows that V is a Lyapunov function, that is, V(x ⁿ⁺¹)≤V(x ⁿ ) along a solution {x ⁿ }^∞ _n=0, where the time steps can be relatively large.