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Biological network dynamics: Poincaré-Lindstedt series and the effect of delays

Renato Calleja, Pablo Padilla-Longoria, Edgar Rodríguez-Mendieta

TL;DR

This work analyzes Hopf bifurcation in a two-delay, non-diffusive Gierer–Meinhardt activator–inhibitor system modeled as a delay differential equation. It proves the existence of a Poincaré–Lindstedt series to all orders by framing the problem in Fourier–Taylor form, yielding explicit linear recurrences for the-series coefficients and enabling implementation via automatic differentiation. The PL series serves as an efficient initial guess for a collocation-based Newton solver, and together with pseudo-arclength continuation the authors compute and continue branches of periodic solutions across delays. The approach provides a practical framework for analytic and numerical tracking of delay-induced periodic dynamics in biological networks, with potential extensions to diffusion terms.

Abstract

This paper focuses on the Hopf bifurcation in an activator-inhibitor system without diffusion which can be modeled as a delay differential equation. The main result of this paper is the existence of the Poincaré-Lindstedt series to all orders for the bifurcating periodic solutions. The model has a non-linearity which is non-polynomial, and yet this allows us to exploit the use of Fourier-Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincaré-Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute branches of periodic solutions.

Biological network dynamics: Poincaré-Lindstedt series and the effect of delays

TL;DR

This work analyzes Hopf bifurcation in a two-delay, non-diffusive Gierer–Meinhardt activator–inhibitor system modeled as a delay differential equation. It proves the existence of a Poincaré–Lindstedt series to all orders by framing the problem in Fourier–Taylor form, yielding explicit linear recurrences for the-series coefficients and enabling implementation via automatic differentiation. The PL series serves as an efficient initial guess for a collocation-based Newton solver, and together with pseudo-arclength continuation the authors compute and continue branches of periodic solutions across delays. The approach provides a practical framework for analytic and numerical tracking of delay-induced periodic dynamics in biological networks, with potential extensions to diffusion terms.

Abstract

This paper focuses on the Hopf bifurcation in an activator-inhibitor system without diffusion which can be modeled as a delay differential equation. The main result of this paper is the existence of the Poincaré-Lindstedt series to all orders for the bifurcating periodic solutions. The model has a non-linearity which is non-polynomial, and yet this allows us to exploit the use of Fourier-Taylor series to develop order-by-order calculations that lead to linear recurrence equations for the coefficients of the Poincaré-Lindstedt series. As applications, we implement the computation of the coefficients of these series for any finite order, and use a pseudo-arclength continuation to compute branches of periodic solutions.

Paper Structure

This paper contains 6 sections, 6 theorems, 90 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Hopf Bifurcation
  3. Poincaré - Lindstedt Series for a periodic solution
  4. Numerical Results
  5. Appendix
  6. Acknowledgment

Key Result

Proposition 2.3

If $b_{1}\neq 0$ and $b_{0}^{2} - b_{2}^{2} < 0$, then system (DDE_GM_around_0) undergoes a Hopf bifurcation, whose bifurcation values are $\gamma_{0}$ given by (gamma0_hopf_bifurcation_b1_neq_0) and $\omega_{0}$ given by (omega0_hopf_bifurcation_b1_neq_0). If $b_{1}=0$ and $b_{2}>1$ then system (DD

Figures (6)

  • Figure 1: Hopf bifurcation points for different Parameter Sets.
  • Figure 2: Different Poincaré-Lindstedt series of periodic solutions $y$ of (\ref{['DDE_GM_around_0']}) and its behavior with particular values of $\varepsilon$.
  • Figure 3: Distance between the Poincaré-Lindstedt at order $\varepsilon^{3}$ and the solution obtained by a Newton method.
  • Figure 4: First branches parameterized by $\gamma$ for the Parameter Set 1. Note the asymptotic behavior in almost all of these branches with the $L_{2}-$norm. These graphs are related with the Figure \ref{['f:most_right_periodic_solutions_Par_set_1']}, which suggests the presence of homoclinic orbits.
  • Figure 5: Continuation of periodic solution of equation (\ref{['DDE_GM_around_0']}) with different parameter sets.
  • ...and 1 more figures

Theorems & Definitions (13)

  • Remark 2.1
  • Proposition 2.3
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Lemma 3.4
  • proof
  • ...and 3 more