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 epsilon 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 epsilon <= delta <= 1. 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 (epsilon, 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.