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An exact solution for a non-autonomous delay differential equation

Kenta Ohira

TL;DR

The paper addresses finding an exact, closed-form solution to the non-autonomous delay differential equation $\frac{dX(t)}{dt}+ a\,t\,X(t)= b\,X(t-\tau)$ over the real line. It develops both a Fourier-transform integral representation and a Gaussian-sum series expansion, valid for $a>0$ and $\tau\neq0$, and shows how negative delays can be incorporated. The main contributions include the first explicit solution for a non-autonomous DDE, a rigorous link between integral and series forms, a stability analysis across parameter regimes, and practical tools for estimating the envelope and maximum of the dynamics. These results offer analytical insight into delay-induced behavior and provide benchmarks for approximations in non-autonomous delay systems with potential applications in physics and engineering.

Abstract

We derive an exact solution for a simple non-autonomous delay differential equation (DDE) over the entire real-time axis, representing it as a sum of Gaussian-shaped dynamics with distinct peak positions. This marks the first explicit solution for non-autonomous DDEs and is a rare example even among general DDEs. The constructed solution offers key physical insights and facilitates the analysis of system properties, such as the envelope profile of the dynamics.

An exact solution for a non-autonomous delay differential equation

TL;DR

The paper addresses finding an exact, closed-form solution to the non-autonomous delay differential equation over the real line. It develops both a Fourier-transform integral representation and a Gaussian-sum series expansion, valid for and , and shows how negative delays can be incorporated. The main contributions include the first explicit solution for a non-autonomous DDE, a rigorous link between integral and series forms, a stability analysis across parameter regimes, and practical tools for estimating the envelope and maximum of the dynamics. These results offer analytical insight into delay-induced behavior and provide benchmarks for approximations in non-autonomous delay systems with potential applications in physics and engineering.

Abstract

We derive an exact solution for a simple non-autonomous delay differential equation (DDE) over the entire real-time axis, representing it as a sum of Gaussian-shaped dynamics with distinct peak positions. This marks the first explicit solution for non-autonomous DDEs and is a rare example even among general DDEs. The constructed solution offers key physical insights and facilitates the analysis of system properties, such as the envelope profile of the dynamics.

Paper Structure

This paper contains 27 sections, 59 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Solution Stability
  3. Case $a \neq 0$
  4. Case $a=0$ and $\tau>0$
  5. Case $a=0$ and $\tau=0$
  6. Analysis Using the Fourier Transform
  7. Fourier Transformability
  8. Case $a<0$:
  9. Case $a=0$ and $\tau\neq0$:
  10. Case $a=0$ and $\tau=0$:
  11. Case $a>0$:
  12. Conclusion:
  13. Derivation of the General Solution in Integral Form
  14. Complex Integration
  15. Series Solution
  16. ...and 12 more sections

Figures (8)

  • Figure 1: The integration path is defined as the rectangle in the complex plane with vertices at $z = -R - i\sqrt{a}t$, $R - i\sqrt{a}t$, $R$, and $-R$. The path on the left corresponds to $t > 0$, while the one on the right corresponds to $t < 0$. The direction of integration is indicated by the arrows.
  • Figure 2: Graph of $t - X(t)$ for the series solution (\ref{['series_expansion_of_the_solution']}) with $a > 0$ and $\tau > 0$, with varying delay $\tau$. We set ${\cal{C}} =1$ and parameters are $a = 0.15$, $b = 6$ with $\tau$ set to (A) 3, (B) 5, (C) 6, (D) 8, (E) 12, (F) 20, (G) 30, (H) 40, and (I) 50. The series sums to $n = 500$.
  • Figure 3: Graph of $t - X(t)$ for the series solution (\ref{['series_expansion_of_the_solution']}) with $a > 0$ and $\tau > 0$, with varying delay $\tau$. We set ${\cal{C}} =1$ and parameters are $a = 0.15$, $b = -6$ with $\tau$ set to (A) 3, (B) 5, (C) 6, (D) 8, (E) 12, (F) 20, (G) 30, (H) 40, and (I) 50. The series sums to $n = 500$.
  • Figure 4: For parameter values $(A)$$a = 0.15, b = -6$ and $(B)$$a = 0.15, b = 6$, with delay $\tau = -8$ in both cases. i: $X(t, b, \tau)$, ii: $X(t, -b, -\tau)$, iii: a comparison of both, showing symmetry.
  • Figure 5: Dynamics of $X(t)$ (blue) and $G(t)$ (orange). $G(t)$ is defined as the curve surrounding the figure. We set ${\cal{C}} =1$ and the parameters are $a = 0.15, b = 6.0$. $\tau$ takes the values (A) 3, (B) 5, (C) 6, (D) 8, (E) 12, (F) 20.
  • ...and 3 more figures