[Back]

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