Connecting orbits for delay differential equations with unimodal feedback

This paper considers a class of delay differential equations with unimodal feedback and describes the structure of certain unstable sets of stationary points and periodic orbits. These unstable sets consist of heteroclinic connections from stationary points and periodic orbits to stable stationary points, stable periodic orbits and some more complicated compact invariant sets. A prototype example is the Mackey--Glass type equation y'(t)=-ay(t)+b y²(t-1)/1+yⁿ(t-1) having three stationary solutions $0$, ξ1,n and ξ2,n with 0<ξ1,n<ξ2,n, provided $b>a>0$, and n is large. The 1-dimensional leading unstable set Wᵘ(ξ1,n) of the stationary point ξ1,n is decomposed into three disjoint orbits, Wᵘ(ξ1,n)=Wu,-(ξ1,n)∪ \ξ1,n\ ∪ Wu,+(ξ1,n).. Here ξ1,n is a constant function in the phase space with value ξ1,n. Wu,-(ξ1,n) is a connecting orbit from ξ1,n to 0̂. There exists a threshold value b^*=b^*(a)>a such that, in case b∈ (a,b^*), Wu,+(ξ1,n) connects ξ1,n to 0̂; and in case b>b^*, Wu,+(ξ1,n) connects ξ1,n to a compact invariant set Aₙ not containing 0̂ and ξ1,n. Under additional conditions, there is a stable periodic orbit Oⁿ with Aₙ= Oⁿ. Analogous results are obtained for the 2-dimensional leading unstable sets Wᵘ(Qⁿ) of periodic orbits Qⁿ close to ξ1,n, establishing connections from Qⁿ to Oⁿ.