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Distributed Delay Systems with Oscillations in the Time Histories

Adam Mielke, Mads Peter Sørensen, John Wyller

TL;DR

This work addresses distributed delay dynamical systems in which the memory kernel exhibits oscillations in time history. It develops a Linear Chain Trick (LCT) that converts such memory terms into finite‑dimensional ODEs and demonstrates the approach on a logistic population model with a perturbed Erlang memory kernel. Through equilibrium and Hopf bifurcation analysis, the authors show that small oscillatory perturbations deform but preserve key dynamical features, and they prove a continuity result linking memory kernels to solution trajectories. The framework is then extended to discuss Raman scattering in nonlinear optics and Kaldor‑Kalecki economic growth models, highlighting broad applicability to systems with oscillatory memory across physics and economics.

Abstract

We design a Linear Chain Trick (LCT)-algorithm for dynamical systems with distributed time delay where the time histories contain temporal oscillations. The methodology is illustrated by means of an example in population dynamics.

Distributed Delay Systems with Oscillations in the Time Histories

TL;DR

This work addresses distributed delay dynamical systems in which the memory kernel exhibits oscillations in time history. It develops a Linear Chain Trick (LCT) that converts such memory terms into finite‑dimensional ODEs and demonstrates the approach on a logistic population model with a perturbed Erlang memory kernel. Through equilibrium and Hopf bifurcation analysis, the authors show that small oscillatory perturbations deform but preserve key dynamical features, and they prove a continuity result linking memory kernels to solution trajectories. The framework is then extended to discuss Raman scattering in nonlinear optics and Kaldor‑Kalecki economic growth models, highlighting broad applicability to systems with oscillatory memory across physics and economics.

Abstract

We design a Linear Chain Trick (LCT)-algorithm for dynamical systems with distributed time delay where the time histories contain temporal oscillations. The methodology is illustrated by means of an example in population dynamics.
Paper Structure (10 sections, 3 theorems, 102 equations, 2 figures)

This paper contains 10 sections, 3 theorems, 102 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Example and Construction of Oscillations in Time History
  3. Linear Chain Trick for systems with oscillating dependence on the time history
  4. Concluding remarks and extensions
  5. Main results
  6. Possible extensions
  7. Raman scattering process
  8. Kaldor-Kalecki economic growth model
  9. Continuous dependence on the time history
  10. The characteristic polynomial (\ref{['characteristicpolynomiallogistic']})

Key Result

Theorem 1

Let $\{\{\delta_{n}^{(k)}\}_{n=1}^{M_{k}}\}_{k=1}^{N}$ be the complex-valued sequence defined by and assume that the component vector $\textbf{x}=\textbf{x}_{e}$ satisfies the condition where and Then $\textbf{X}_{e}$ defined by (equilibriumGsystem) is an equilibrium point of the dynamical system $\dot{\textbf{X}}=\textbf{G}(\textbf{X})$.

Figures (2)

  • Figure 1: Phase diagram of the logistic equation with oscillations in the time history for different values of $\varepsilon$ and $\sigma$. The input parameters are $r=2$, $K=1$, and $\Omega=0.8$. The critical line satisfies (\ref{['GenHopf']}) as well as $|\mathbf{D}_j|>0$ for $j=1\dots 5$.
  • Figure 2: Numerical solution of the logistic equation with oscillations in the time history for different values of $\varepsilon$ and $\sigma$. That is, the system of equations (\ref{['rate1a']}), (\ref{['rate2']}) and (\ref{['logisticdelay']}). The input parameters are $r=2$, $K=1$, and $\Omega=0.8$. The combinations of $\varepsilon$ and $\sigma$ are chosen along the critical line in Figure \ref{['Fig:PhaseDiagramLogistic']}, and the background color is based on that figure as well. Note how the oscillations of all state variables increase in magnitude in the unstable regime, but decrease in the stable one.

Theorems & Definitions (14)

  • Remark 1
  • Remark 2
  • Theorem 1
  • proof
  • Remark 3
  • Remark 4
  • Remark 5
  • Remark 6
  • Lemma 1
  • proof
  • ...and 4 more