Existence, uniqueness, and stability of slowly oscillating periodic solutions for delay differential equations with nonnegativity constraints

David Lipshutz and Ruth J. Williams

Deterministic dynamical system models with delayed feedback and nonnegativity constraints arise in a variety of applications in science and engineering. Under certain conditions oscillatory behavior has been observed and it is of interest to know when this behavior is periodic. Here we consider one-dimensional delay differential equations with nonnegativity constraints as prototypes for such models. We obtain sufficient conditions for the existence of slowly oscillating periodic solutions (SOPS) of such equations when the delay/lag interval is long and the dynamics depend only on the current and delayed state. Under further assumptions, including possibly longer delay intervals and restricting the dynamics to depend only on the delayed state, we prove uniqueness and exponential stability for such solutions. To prove these results, we develop a theory for studying perturbations of these constrained SOPS. We illustrate our results with simple examples of biochemical reaction network models and an Internet rate control model.

Appeared in SIAM J. Math. Anal., Vol. 47, No. 6 (2015), pp. 4467-4535. For a copy of a reprint of the paper, click here.

Note that there is a typo on page 4474: in the sentence before section 3.2, $q_0=\tau$ should be $q_0=-\tau$, i.e., a minus sign is missing.