A Multiparameter Singular Perturbation Analysis of the Robertson Model

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 $({\epsilon }_1,{\epsilon }_2):=(k_1/k_2,k_3/k_2)$, which we then analyze using geometric singular perturbation theory (GSPT). To deal with the multiparameter singular structure, we perform blowups in parameter- and variable space. We identify four distinct regimes in a neighborhood of the singular limit $({\epsilon }_1,{\epsilon }_2)=(0,0)$. Within these four regimes, we use GSPT and additional blowups to analyze the dynamics and the structure of solutions. Our asymptotic results are in excellent qualitative and quantitative agreement with the numerics.