Scaling in Singular Perturbation Problems: Blowing Up a Relaxation Oscillator

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 ε and δ. In [20] it was argued that a limit cycle of relaxation type exists for ε ¿ δ ¿ 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 (ε, δ) = (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.