Existence, uniqueness and stability of slowly oscillating periodic solutions for delay differential equations with non-negativity constraints
D. Lipshutz and
R. J. Williams
Deterministic dynamical system models with delayed feedback and non-negativity 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 non-negativity 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 genetic circuit models and an Internet rate control model.
Appeared in SIAM Journal on Mathematical Analysis, Vol. 47 (2015), 4467-4535.
For a copy of a preprint of the Lipshutz-Williams paper for personal, scientific, non commercial use,
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Note. There is a typo on page 4474: in the sentence before section 3.2, tau
should be -tau,
i.e., a minus sign is missing.