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A Multi-Parameter Singular Perturbation Analysis of the Robertson Model

Lukas Baumgartner, Peter Szmolyan

TL;DR

The Robertson model is analyzed as a two-parameter singular perturbation problem with $\varepsilon_1=k_1/k_2$ and $\varepsilon_2=k_3/k_2$. By employing geometric singular perturbation theory together with blow-up in parameter and phase space, the authors reveal four distinct slow–fast regimes near $(\varepsilon_1,\varepsilon_2)=(0,0)$ and construct singular orbits that connect the initial state $O$ to the equilibrium $Q$, which persist as the parameters become small. The main contributions include a systematic multi-parameter blow-up framework, regime-specific slow–fast analyses, and rigorous existence results for perturbed orbits $\gamma_r$, with quantitative predictions such as $y^{\max}=\sqrt{\varepsilon_1 c}+\mathcal{O}(\varepsilon_1)$ (and refinements in other regions). The analytical results show excellent qualitative and quantitative agreement with numerical simulations and provide a versatile template for multi-parameter singular perturbations in chemical kinetics and related biological models.

Abstract

The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates $k_1,\,k_2$, and $k_3$, with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption $k_1, k_3 \ll k_2$. We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters $(\varepsilon_1,\varepsilon_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit $(\varepsilon_1,\varepsilon_2)= (0,0)$. Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics.

A Multi-Parameter Singular Perturbation Analysis of the Robertson Model

TL;DR

The Robertson model is analyzed as a two-parameter singular perturbation problem with and . By employing geometric singular perturbation theory together with blow-up in parameter and phase space, the authors reveal four distinct slow–fast regimes near and construct singular orbits that connect the initial state to the equilibrium , which persist as the parameters become small. The main contributions include a systematic multi-parameter blow-up framework, regime-specific slow–fast analyses, and rigorous existence results for perturbed orbits , with quantitative predictions such as (and refinements in other regions). The analytical results show excellent qualitative and quantitative agreement with numerical simulations and provide a versatile template for multi-parameter singular perturbations in chemical kinetics and related biological models.

Abstract

The Robertson model describing a chemical reaction involving three reactants is one of the classical examples of stiffness in ODEs. The stiffness is caused by the occurrence of three reaction rates , and , with largely differing orders of magnitude, acting as parameters. The model has been widely used as a numerical test problem. Surprisingly, no asymptotic analysis of this multiscale problem seems to exist. In this paper we provide a full asymptotic analysis of the Robertson model under the assumption . We rewrite the equations as a two-parameter singular perturbation problem in the rescaled small parameters , which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multi-parameter singular structure, we perform blow-ups in parameter- and variable space. We identify four distinct regimes in a neighbourhood of the singular limit . Within these four regimes we use GSPT and additional blow-ups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics.
Paper Structure (8 sections, 13 theorems, 108 equations, 14 figures, 1 table)

This paper contains 8 sections, 13 theorems, 108 equations, 14 figures, 1 table.

Table of Contents

  1. Introduction
  2. Structure of Parameter Space
  3. Analysis in Region $B_2$
  4. Analysis in Region $B_1$
  5. Analysis in region $B_{11}$
  6. Analysis in region $B_{12}$
  7. Analysis in Region $B_3$
  8. Summary and Outlook

Key Result

Lemma 1.1

All solutions of equ:rob_reaction starting in the non-negative orthant $\mathbb{R}^3_+$ exist globally in forward time. The $z$-axis is a line of attracting equilibria. The solution with initial value $(x_0,y_0,z_0)^T \in \mathbb{R}^3_+$ converges to the equilibrium $(\hat{x},\hat{y},\hat{z})^T=(0,0

Figures (14)

  • Figure 1: Reaction scheme of the Robertson model.
  • Figure 2: Numerical solution of equation \ref{['equ:rob_reaction']} with an implicit BDF solver Scipy_BDF. Note the logarithmic time scale.
  • Figure 3: The four scaling regions $B_{11}$, $B_{12}$, $B_2$, and $B_3$.
  • Figure 4: Numerical simulations of \ref{['equ:rob_reaction_2D']} in different regions of the parameter space for $c=1$.
  • Figure 5: Parameter blow-up $\Phi_{par}^1$ of the origin and charts $\mathcal{P}_1$ (orange) and $\mathcal{P}_2$ (blue)
  • ...and 9 more figures

Theorems & Definitions (39)

  • Lemma 1.1
  • proof
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Theorem 1.5
  • Remark 1.6
  • Remark 2.1
  • Lemma 3.1
  • Theorem 3.2
  • ...and 29 more