Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback

Abstract

We consider delay differential equations of the form y′(t)=-ay(t)+bf(y(t-1)) with positive parameters a, b and a unimodal f:[0,∞)→[0,1]. It is assumed that the nonlinear f is close to a function g:[0,∞)→[0,1] with g(ξ)=0 for all ξ>1. The fact g(ξ)=0 for all ξ>1 allows to construct stable periodic orbits for the equation x′(t)=-cx(t)+dg(x(t-1)) with some parameters d>c>0. Then it is shown that the equation y′(t)=-ay(t)+bf(y(t-1)) also has a stable periodic orbit provided a, b, f are sufficiently close to c, d, g in a certain sense. The examples include f(ξ)=ξk1+ξn for parameters k>0 and n>0 together with the discontinuous g(ξ)=ξk for ξ∈[0,1), and g(ξ)=0 for ξ>1. The case k=1 is the famous Mackey–Glass equation, the case k>1 appears in population models with Allee effect, and the case k∈(0,1) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.