Families of periodic delay orbits

TL;DR

This work analyzes delay differential equations in the plane by constructing explicit families of 1-periodic delay orbits parameterized by the delay $\tau$. It represents the dependence of these orbits on $\tau$ as a plane curve $\gamma_{\alpha,\rho}(\tau)=z_{\alpha,\rho,\tau}(0)$ and shows that, in the non-degenerate setting, these curves are smooth except at cusps, where the tangent direction reverses. The authors develop a systematic framework for gluing local delay-family branches at degeneracies and illustrate the constructions with concrete examples using both trigonometric and polynomial $f$ and $g$, including explicit formulas and numerical visuals. The results provide explicit, highly structured delay-orbit families in ${\mathbb R}^2$, clarifying how delay interacts with radial and angular dynamics and offering practical methods to build smooth, connected families across degeneracies. This advances constructive understanding of delay phenomena in low-dimensional dynamics and offers templates for explicit computation and visualization.

Abstract

We construct and analyze families of periodic delay orbits for a class of delay differential equations in two dimensions depending on two real-valued functions. These families are parametrized by the delay parameter. It is possible to represent the dependency of these periodic delay orbits on the delay parameter by a curve in the plane, without loss of information. It turns out that the singularities of these curves necessarily are cusps in the non-degenerate case. After discussing degenerate situations in general, we explain how to glue different families of periodic delay orbits at degeneracies in the delay parameter.

Figures (11)

• Figure 1: (Normalized) Exemplary vector fields $X$ at different times $f(\theta)=2\cdot \cos(10 \pi \theta) \cdot \sin (4 \pi \theta)$$g(r) = e^r\cdot \sin(2 \pi r)$
• Figure 2: Construction of $t_{\alpha,\tau}$ and $r_{\rho,\tau}$ for monotone $f$ and $\widetilde{g}$
• Figure 3: Families of delay orbits for the example in Figure \ref{['fig:example-vfield-a']}. The figures show the smooth $S^1$ families we constructed in the discussion above. Moreover, the role and explicit choice of $\alpha$ and $\rho$ is depicted to show their influence on the delay families.
• Figure 4: On the left we have an example for a cusp due to condition (i) and on the right for condition (ii) of Lemma 4.2.
• Figure 5: Examples of delay families with two cusps

Theorems & Definitions (21)

• Theorem 1.1: AlbersSeifert22
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Definition 2.3
• Lemma 2.4: AlbersSeifert22
• Lemma 2.5
• proof
• Lemma 3.1