Connecting orbits for delay differential equations with unimodal feedback

TL;DR

The paper analyzes delay differential equations with unimodal feedback, focusing on the structure of unstable sets of equilibria and nearby periodic orbits. By combining monotone semiflow theory with a closeness framework that relates a discontinuous limiting model to smooth approximations, it establishes precise descriptions of connecting orbits and the emergence of stable periodic orbits, including a critical threshold $d^*(c)$. For large parameter n, the smooth approximations inherit stable periodic orbits $\mathcal{O}^n$ and exhibit leading unstable sets that connect to $0$ or to $\mathcal{O}^n$, while a Hopf-based construction yields additional periodic orbits $q^n$ and their corresponding unstable manifolds. Together, the results provide a rigorous geometric picture of how unimodal delay feedback shapes global dynamics, including structured heteroclinic connections and parameter regimes compatible with complex transient behavior. The work links a limiting discontinuous feedback to smooth models and clarifies when robust periodic solutions arise, contributing to the rigorous understanding of Mackey–Glass–type systems and their bifurcations.

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 $0$, $ξ_{1,n}$ and $ξ_{2,n}$ with $0<ξ_{1,n}<ξ_{2,n}$, provided $b>a>0$, and $n$ is large. The 1-dimensional leading unstable set $W^u(\hatξ_{1,n})$ of the stationary point $\hatξ_{1,n}$ is decomposed into three disjoint orbits, $W^u(\hatξ_{1,n})=W^{u,-}(\hatξ_{1,n})\cup \{\hatξ_{1,n}\} \cup W^{u,+}(\hatξ_{1,n})$.. Here $\hatξ_{1,n}$ is a constant function in the phase space with value $ξ_{1,n}$. $W^{u,-}(\hatξ_{1,n})$ is a connecting orbit from $\hatξ_{1,n}$ to $\hat{0}$. There exists a threshold value $b^*=b^*(a)>a$ such that, in case $b\in (a,b^*)$, $W^{u,+}(\hatξ_{1,n})$ connects $\hatξ_{1,n}$ to $\hat{0}$; and in case $b>b^*$, $W^{u,+}(\hatξ_{1,n})$ connects $\hatξ_{1,n}$ to a compact invariant set $\mathcal{A}_n$ not containing $\hat{0}$ and $\hatξ_{1,n}$. 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 $\hatξ_{1,n}$, establishing connections from $\mathcal{Q}^n$ to $\mathcal{O}^n$.

Figures (5)

• Figure 1: Visualization of the results of Theorems \ref{['thm:Efn']} and \ref{['thm:Hopf']}
• Figure 2: The case $a=1$, $b=7.38$, $n$ large.
• Figure 3: The case $a=4$, $b=12.71$, $n$ large. The structure of the periodic solution looks complicated, although the corresponding periodic orbit $\mathcal{O}^n$ is stable.
• Figure 4: The case $a\approx \frac{5\pi}{3\sqrt{3}}$, $b\approx 25$, $n$ large.
• Figure 5: The case $a\approx \frac{5\pi}{3\sqrt{3}}$, $b\approx 7.95$, $n$ large.

Theorems & Definitions (47)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• proof
• Proposition 2.2
• Proposition 2.3
• proof
• Proposition 2.4
• proof