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≪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→0, so that in polar coordinates (θ,r), and by doing a further change of variable r↦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.