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Universal bifurcation scenarios in delay-differential equations with one delay

Yu Wang, Jinde Cao, Jürgen Kurths, Serhiy Yanchuk

TL;DR

The paper identifies universal bifurcation scenarios for delay-differential equations with a single delay by classifying linear ACS shapes into three main universality classes (I–III) and linking them to delay-induced Hopf crossings through a transversality theorem. It introduces the generating polynomial $p_\omega(Y)$ and the ACS branches $\gamma_j(\omega)$ to predict critical delays $\tau_k=\frac{1}{\omega_H}(\phi_H+2\pi k)$ and the direction of bifurcations. The authors provide explicit results for scalar and two-variable DDEs, with illustrative examples, and show how these universal cascades extend to nonlinear dynamics, exemplified by a Stuart–Landau oscillator with time-delayed feedback. The framework clarifies how delay parameters organize stability and periodic solutions across many systems, offering a practical tool for anticipating and controlling delay-induced dynamics.

Abstract

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart-Landau oscillator with time-delayed feedback.

Universal bifurcation scenarios in delay-differential equations with one delay

TL;DR

The paper identifies universal bifurcation scenarios for delay-differential equations with a single delay by classifying linear ACS shapes into three main universality classes (I–III) and linking them to delay-induced Hopf crossings through a transversality theorem. It introduces the generating polynomial and the ACS branches to predict critical delays and the direction of bifurcations. The authors provide explicit results for scalar and two-variable DDEs, with illustrative examples, and show how these universal cascades extend to nonlinear dynamics, exemplified by a Stuart–Landau oscillator with time-delayed feedback. The framework clarifies how delay parameters organize stability and periodic solutions across many systems, offering a practical tool for anticipating and controlling delay-induced dynamics.

Abstract

We show that delay-differential equations (DDE) exhibit universal bifurcation scenarios, which are observed in large classes of DDEs with a single delay. Each such universality class has the same sequence of stabilizing or destabilizing Hopf bifurcations. These bifurcation sequences and universality classes can be explicitly described by using the asymptotic continuous spectrum for DDEs with large delays. Here, we mainly study linear DDEs, provide a general transversality result for the delay-induced bifurcations, and consider three most common universality classes. For each of them, we explicitly describe the sequence of stabilizing and destabilizing bifurcations. We also illustrate the implications for a nonlinear Stuart-Landau oscillator with time-delayed feedback.
Paper Structure (21 sections, 2 theorems, 117 equations, 16 figures)

This paper contains 21 sections, 2 theorems, 117 equations, 16 figures.

Table of Contents

  1. Introduction
  2. Model and notations
  3. Critical characteristic roots and transversality theorem
  4. Universality class I
  5. Scalar DDEs
  6. Two-variable linear DDE
  7. Example for the case .
  8. Example for the case
  9. Universality class II
  10. First example for universality class II
  11. Second example for universality class II
  12. Universality class III
  13. Two-variable example of DDE of universality class III with .
  14. Two-variable example of DDE of universality class III with .
  15. Nonlinear model example: Stuart-Landau oscillator with time-delayed feedback
  16. ...and 6 more sections

Key Result

Theorem 3.1

Let $\lambda_c=i\omega_H$$(\omega_H>0)$ be a critical simple characteristic root of the linear DDE eq:DDE for $\tau=\tau_H$. Then there is a delay-induced transverse crossing of this characteristic root at $\tau=\tau_H$ if and only if there is a branch of ACS with $\gamma(\omega_H)=0$ and $\frac{d}{

Figures (16)

  • Figure 1: Schematic representation of class I asymptotic continuous spectrum (ACS). $\lambda=\pm i\omega_H$ are the only possible critical characteristic roots.
  • Figure 2: Asymptotic continuous spectrum for the scalar DDE \ref{['eq:ce-scalar']}.
  • Figure 3: The spectrum of the scalar DDE \ref{['eq:DDE-scalar']} with $a=-0.5$, $b=-1$, and (a) $\tau=\tau_0=2.4184$; (b) $\tau=\tau_1=9.6736$; (c) $\tau=\tau_8=60.4600$, where $\tau_k$ is given by \ref{['eq:tauk_scalar']}. The dashed line denotes the curve of the asymptotic continuous spectrum. The case (a) corresponds to the first pair or eigenvalues becomes critical, (b) the second pair, and (c) to the case when 8 pairs of eigenvalues are unstable and the 9th pair is critical. The critical frequency and the phase are $\omega_{H}=0.866$ and $\phi_{H}=2.0944$.
  • Figure 4: The spectrum of the two-variable delay system with the parameters \ref{['eq:AB']} and the delay $\tau_6 \approx 28.6$. The dashed line denotes the curve of the asymptotic continuous spectrum.
  • Figure 5: Example of the spectrum for the two-variable delay system \ref{['eq:TwovarDDE']} with $\det(B)\ne 0$. The coefficients are given in \ref{['eq:ABa']} for (a) and \ref{['eq:ABb']} for (b). Delay values are (a) $\tau=\tau_9=13.091$, (b) $\tau=\tau_{10}=32.06$. The spectrum in (a) corresponds to a negative value of $C$ (see Eq. \ref{['eq:C']}) and (b) to a positive value of $C$.
  • ...and 11 more figures

Theorems & Definitions (20)

  • definition 1: Delay-induced transverse crossing
  • definition 2: Stabilizing and destabilizing transverse crossing
  • definition 3: Generating polynomial
  • definition 4: Asymptotic Continuous Spectrum, ACS
  • Theorem 3.1: Transversality Theorem
  • proof
  • Theorem 3.2: Universal sequence of crossing events / Bifurcation Theorem
  • proof
  • definition 5: Class I asymptotic spectrum
  • definition 6: Class I DDEs
  • ...and 10 more