Stable Periodic Orbits for Delay Differential Equations with Unimodal Feedback

Abstract

AbstractWe consider delay differential equations of the form $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( y ( t - 1 ) ) with positive parameters a, b and a unimodal $$f:[0,\infty )\rightarrow [0,1]$$ f : [ 0 , ∞ ) → [ 0 , 1 ] . It is assumed that the nonlinear f is close to a function $$g:[0,\infty )\rightarrow [0,1]$$ g : [ 0 , ∞ ) → [ 0 , 1 ] with $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 . The fact $$g(\xi )=0$$ g ( ξ ) = 0 for all $$\xi >1$$ ξ > 1 allows to construct stable periodic orbits for the equation $$x^{\prime }(t)=-cx(t)+dg(x(t-1))$$ x ′ ( t ) = - c x ( t ) + d g ( x ( t - 1 ) ) with some parameters $$d>c>0$$ d > c > 0 . Then it is shown that the equation $$ y^{\prime }(t)=-ay(t)+bf(y(t-1)) $$ y ′ ( t ) = - a y ( t ) + b f ( 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 $$f(\xi )=\frac{\xi ^k}{1+\xi ^n}$$ f ( ξ ) = ξ k 1 + ξ n for parameters $$k>0$$ k > 0 and $$n>0$$ n > 0 together with the discontinuous $$g(\xi )=\xi ^k$$ g ( ξ ) = ξ k for $$\xi \in [0,1)$$ ξ ∈ [ 0 , 1 ) , and $$g(\xi )=0$$ g ( ξ ) = 0 for $$\xi >1$$ ξ > 1 . The case $$k=1$$ k = 1 is the famous Mackey–Glass equation, the case $$k>1$$ k > 1 appears in population models with Allee effect, and the case $$k\in (0,1)$$ k ∈ ( 0 , 1 ) arises in some economic growth models. The obtained stable periodic orbits may have complicated structures.