Tikhonov's theorem andquasi-steady state

Lena Noethen; Sebastian Walcher

doi:10.3934/dcdsb.2011.16.945

Article Contents

2011, Volume 16, Issue 3: 945-961. Doi: 10.3934/dcdsb.2011.16.945

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Tikhonov's theorem and quasi-steady state

Lena Noethen^1, and
Sebastian Walcher^1,

1.
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen

Received: June 30, 2010

Revised: February 28, 2011

Published: September 2011

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Abstract

There exists a systematic approach to asymptotic properties for quasi-steady state phenomena via the classical theory of Tikhonov and Fenichel. This observation allows, on the one hand, to settle convergence issues, which are far from trivial in asymptotic expansions. On the other hand, even if one takes convergence for granted, the approach yields a natural way to compute a reduced system on the slow manifold, with a reduced equation that is frequently simpler than the one obtained by the ad hoc approach. In particular, the reduced system is always rational. The paper includes a discussion of necessary and sufficient conditions for applicability of Tikhonov's and Fenichel's theorems, computational issues and a direct determination of the reduced system. The results are applied to several relevant examples.

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Mathematics Subject Classification: Primary: 34E15, 92C45; Secondary: 80A30.

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