Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator

A detailed geometric analysis of the Goldbeter–Lefever model of glycolytic oscillations is given. 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 [L. Segel and A. Goldbeter, J. Math. Biol., 32 (1994), pp. 147–160] it was argued that a limit cycle of relaxation type exists for $\varepsilon\ll\delta\ll1$. The existence of this limit cycle is proved 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 a novel variant of the blow-up method. It is shown that repeated blow-ups lead to a clear geometric picture of this fairly complicated two-parameter multiscale problem.