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Scaling in singular perturbation problems: blowing-up a relaxation oscillator
Ilona Gucwa and Peter Szmolyan
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 $\varepsilon$ 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 $\varepsilon \ll \delta \ll 1$. 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 $(\varepsilon,\delta) = (0,0)$ 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.