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Calculation of explicit expressions for the Hopf bifurcation limit cycles in delay-differential equations

José Enríquez Gabeiras, Juan Franciasco Padial Molina

TL;DR

The paper develops a systematic extension of the Casal–Freedman Poincaré-Lindstedt framework to compute explicit $2\pi$-periodic Hopf limit cycles in autonomous delay differential equations with a single fixed delay $\lambda$, producing uniformly valid asymptotic power-series in the small parameter $\varepsilon$. It introduces a detailed, algorithmic procedure to solve a hierarchy of linear DDEs for each order $j$, enforcing solvability via adjoint orthogonality and assembling the complete periodic solution $\mathbf{Z}(\tau,\varepsilon)$ to recover the original state. The methodology is demonstrated on two representative models: a car-following delay model (nDDE) and a SIR epidemic model with temporary immunity, delivering explicit expressions for the periodic solutions, their frequency, and period corrections, together with rigorous residual-based error estimates. The results enable rapid, high-precision bifurcation diagrams and explicit insight into how delays influence oscillation amplitude and frequency, with potential applicability to higher-dimensional Hopf bifurcations and more complex DDEs. The approach provides a practical tool for точно characterizing delay-induced oscillations in applications where explicit expressions are valuable for analysis, control, or SEO-driven summaries.

Abstract

This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980 by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to two relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model, whose oscillations are a plausible explanation for the density waves and congestion in road traffic, and a SIR epidemic model for propagation of diseases with temporary immunity.

Calculation of explicit expressions for the Hopf bifurcation limit cycles in delay-differential equations

TL;DR

The paper develops a systematic extension of the Casal–Freedman Poincaré-Lindstedt framework to compute explicit -periodic Hopf limit cycles in autonomous delay differential equations with a single fixed delay , producing uniformly valid asymptotic power-series in the small parameter . It introduces a detailed, algorithmic procedure to solve a hierarchy of linear DDEs for each order , enforcing solvability via adjoint orthogonality and assembling the complete periodic solution to recover the original state. The methodology is demonstrated on two representative models: a car-following delay model (nDDE) and a SIR epidemic model with temporary immunity, delivering explicit expressions for the periodic solutions, their frequency, and period corrections, together with rigorous residual-based error estimates. The results enable rapid, high-precision bifurcation diagrams and explicit insight into how delays influence oscillation amplitude and frequency, with potential applicability to higher-dimensional Hopf bifurcations and more complex DDEs. The approach provides a practical tool for точно characterizing delay-induced oscillations in applications where explicit expressions are valuable for analysis, control, or SEO-driven summaries.

Abstract

This paper introduces a methodology to derive explicit power series approximations for the limit cycle periodic solutions of the Hopf bifurcation in autonomous discrete delay differential equations (DDE). The procedure extends the methodology introduced by Casal and Freedman in 1980 by providing a detailed algorithm that iteratively performs systematic calculations up to any desired order of approximation, ensuring a specific error tolerance for any nonlinear DDE presenting a Hopf bifurcation. The methodology is applied to two relevant delay-differential models to illustrate its features: a recently introduced car-following mobility model, whose oscillations are a plausible explanation for the density waves and congestion in road traffic, and a SIR epidemic model for propagation of diseases with temporary immunity.

Paper Structure

This paper contains 21 sections, 3 theorems, 78 equations, 7 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Poincaré-Lindstedt method applied to DDEs
  3. System of equations transformation
  4. Solutions as series expansion
  5. Equation for order $j$ of $\varepsilon$
  6. Asymptotic validity of the solution
  7. Procedure to obtain a explicit solution of a desired order
  8. General expression for the solution of order $j$
  9. Solution $\mathbf{Z}_{j} (\tau)$ of $DDE_{Z_{j}}$ \ref{['ecu25']}
  10. Complete solution
  11. Error estimation
  12. Numerical examples
  13. A car following model for road traffic analysis
  14. System of equations and equilibrium solution
  15. Solution bases
  16. ...and 6 more sections

Key Result

Theorem 2.1

Under Assumption A1 and A2, for an $\varepsilon$ small enough, a sufficient condition for the system of equations ecu12, ecu13 and ecu16 to possess a unique and smooth solution $\left( \hat{\lambda}(\varepsilon), \hat{T}(\varepsilon), \mathbf{Z}(\tau, \varepsilon) \right)$, with $\mathbf{Z}(\cdot,

Figures (7)

  • Figure 1: Bifurcation diagram of $s$ and $s'$ in the nDDE model \ref{['ecu83']}.
  • Figure 2: Asymptotic stable solution of $x_{1}$ and $x_{2}$ for $\lambda=1.4s$ in nDDE\ref{['ecu84']}.
  • Figure 3: Asymptotic stable solution of $x_{1}(t)$ and $x_{2}(t)$ for $\lambda=1.6s$ in nDDE\ref{['ecu84']}.
  • Figure 4: Asymptotic stable solution of $x_{1}(t)$ and $x_{2}(t)$ for $\lambda=1.8s$ in nDDE\ref{['ecu84']}.
  • Figure 5: Bifurcation diagram of $I$, $R$ and $S$ in the $SIR$ model \ref{['ecu90']}.
  • ...and 2 more figures

Theorems & Definitions (8)

  • Remark
  • Definition 1
  • Remark
  • Theorem 2.1: Casal, Freedman Casal1980
  • Proposition 3.1
  • proof
  • Corollary
  • proof