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MiS Preprint

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.

Aug 17, 2009
Aug 18, 2009
MSC Codes:
34C26, 34E15, 37C10, 37C27
slow-fast dynamics, relaxation oscillations, geometric singular perturbation theory, blow-up method, slow manifolds

Related publications

2011 Repository Open Access
Ilona Kosiuk and Peter Szmolyan

Scaling in singular perturbation problems : blowing-up a relaxation oscillator

In: SIAM journal on applied dynamical systems, 10 (2011) 4, pp. 1307-1343