A singular perturbation analysis for the Brusselator

Abstract

In this work we study the Brusselator – a prototypical model for chemical oscillations– under the assump- tion that the bifurcation parameter is of order O(1/ϵ) for positive ϵ ≪ 1. The dynamics of this mathematical model exhibits a time scale separation visible via fast and slow regimes along its unique attracting limit cy- cle. This limit cycle accumulates at infinity as ϵ → 0, so that appropriate coordinates (w, z) are used to analyse the dynamics near the line at infinity, corresponding to the set {z = 0}. This object becomes a non- hyperbolic 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 for small ϵ.