I. Gucwa, P. Szmolyan:

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

in: "ASC Report 39/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

An introduction to some recently developed methods for the analysis of systems

of singularly perturbed ordinary di®erential equations is given in the context of a speci¯c problem

describing glycolytic oscillations. In suitably scaled variables the governing equations are a planar

system of ordinary di®erential equations depending singularly on two small parameters " and ±. In

[20] it was argued that a limit cycle of relaxation type exists for " ¿ ± ¿ 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 ("; ±) = (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.

slow-fast dynamics, relaxation oscillations, geometric singular perturbation theory,

http://www.asc.tuwien.ac.at/preprint/2009/asc39x2009.pdf

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