Connecting orbits for delay differential equations with unimodal feedback

Abstract

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 $y'(t)=-ay(t)+b \frac{y^2(t-1)}{1+y^n(t-1)}$ having three stationary solutions $0$, $\xi_{1,n}$ and $\xi_{2,n}$ with $0<\xi_{1,n}<\xi_{2,n}$, provided $b>a>0$, and $n$ is large. The 1-dimensional leading unstable set $W^u(\hat{\xi}_{1,n})$ of the stationary point $\hat{\xi}_{1,n}$ is decomposed into three disjoint orbits, $W^u(\hat{\xi}_{1,n})=W^{u,-}(\hat{\xi}_{1,n})\cup \{\hat{\xi}_{1,n}\} \cup W^{u,+}(\hat{\xi}_{1,n})$.. Here $\hat{\xi}_{1,n}$ is a constant function in the phase space with value $\xi_{1,n}$. $W^{u,-}(\hat{\xi}_{1,n})$ is a connecting orbit from $\hat{\xi}_{1,n}$ to $\hat{0}$. There exists a threshold value $b^*=b^*(a)>a$ such that, in case $b\in (a,b^*)$, $W^{u,+}(\hat{\xi}_{1,n})$ connects $\hat{\xi}_{1,n}$ to $\hat{0}$; and in case $b>b^*$, $W^{u,+}(\hat{\xi}_{1,n})$ connects $\hat{\xi}_{1,n}$ to a compact invariant set $\mathcal{A}_n$ not containing $\hat{0}$ and $\hat{\xi}_{1,n}$. Under additional conditions, there is a stable periodic orbit $\mathcal{O}^n$ with $\mathcal{A}_n= \mathcal{O}^n$. Analogous results are obtained for the 2-dimensional leading unstable sets $W^u(\mathcal{Q}^n)$ of periodic orbits $\mathcal{Q}^n$ close to $\hat{\xi}_{1,n}$, establishing connections from $\mathcal{Q}^n$ to $\mathcal{O}^n$.