MiS Preprint Repository

An introduction to some recently developed methods for the analysis of systems of singularly perturbed ordinary differential equations is given in the context of a specific problem describing glycolytic oscillations. In suitably scaled variables the governing equations are a planar system of ordinary differential equations depending singularly on two small parameters $\epsilon$ and $\delta$. In a previous work on glycolytic oscillations by Segel and Goldbeter "Scaling in biochemical kinetics: dissection of a relaxation oscillator", J. Math. Biol. 32, 147-160 (1994), it was argued that a limit cycle of relaxation type exists for $\epsilon \ll \delta \ll 1$. The existence of this limit cycle is proven by analyzing the problem in the spirit of geometric singular perturbation theory. The degeneracies of the limiting problem corresponding to $(\epsilon ,\delta )=(0,0)$ are resolved by repeatedly applying the blow-up method. It is shown that the blow-up method leads to a clear geometric picture of this fairly complicated two parameter multi-scale problem.