A singular perturbation analysis for the Brusselator

Abstract

In this work we study the Brusselator - a prototypical model for chemical oscillations - under the assumption that the bifurcation parameter is of order $O(1/ε)$ for positive $ε\ll 1$. The dynamics of this mathematical model exhibits a time scale separation visible via fast and slow regimes along its unique attracting limit cycle. Noticeably this limit cycle accumulates at infinity as $ε\rightarrow 0$, so that in polar coordinates $(θ,r)$, and by doing a further change of variable $r\mapsto r^{-1}$, we analyse the dynamics near the line at infinity, corresponding to the set $\{r=0\}$. This object becomes a nonhyperbolic invariant manifold for which we use a desingularising rescaling, in order to study the closeby dynamics. Further use of geometric singular perturbation techniques allows us to give a decomposition of the Brusselator limit cycle in terms of four different fully quantified time scales.