Export file:

Format

  • RIS(for EndNote,Reference Manager,ProCite)
  • BibTex
  • Text

Content

  • Citation Only
  • Citation and Abstract

Coordinate-independent singular perturbation reduction for systems with three time scales

Mathematik A, RWTH Aachen, 52056 Aachen, Germany

Special Issues: Mathematical analysis of reaction networks: theoretical advances and applications

On the basis of recent work by Cardin and Teixeira on ordinary differential equations with more than two time scales, we devise a coordinate-independent reduction for systems with three time scales; thus no a priori separation of variables into fast, slow etc. is required. Moreover we consider arbitrary parameter dependent systems and extend earlier work on Tikhonov-Fenichel parameter values – i.e. parameter values from which singularly perturbed systems emanate upon small perturbations – to the three time-scale setting. We apply our results to two standard systems from biochemistry.
  Figure/Table
  Supplementary
  Article Metrics

References

1. A. N. Tikhonov, Systems of differential equations containing a small parameter multiplying the derivative (in Russian), Math. Sb, 31 (1952), 575–586.

2. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Differ. Equations, 31 (1979), 53–98.

3. H. G. Kaper and T. J. Kaper, Asymptotic analysis of two reduction methods for systems of chemical reactions, Physica D, 165 (2002), 66–93.

4. V. Noel, D. Grigoriev, S. Vakulenko, et al., Tropicalization and tropical equilibrium of chemical reactions, In: G.L. Litvinov, and S.N. Sergeev (eds): Tropical and idempotent mathematics and applications, Contemporary Math., 616, 261–275. Amer. Math. Soc., Providence, 2014.

5. O. Radulescu, S. Vakulenko and D. Grigoriev, Model reduction of biochemical reactions networks by tropical analysis methods, Math. Model. Nat. Phenom., 10 (2015), 124–138.

6. S. S. Samal, D. Grigoriev, H. Fröhlich, et al., Analysis of reaction network systems using tropical geometry, In: V.P. Gerdt, W. Koepf, W.M. Seiler and E.V. Vorozhtsov (eds.): Computer Algebra in Scientific Computing. 17Mth International Workshop, CASC 2015., 424–439, Lecture Notes in Computer Science, 9301, Springer-Verlag, Cham, 2015.

7. S. S. Samal, D. Grigoriev, H. Fröhlich, et al., A geometric method for model reduction of biochemical networks with polynomial rate functions, Bull. Math. Biol., 77 (2015), 2180–2211.

8. C. Chicone, Ordinary differential equations with applications, 2nd edition, Springer-Verlag, New York, 2006.

9. D. Capelletti and C. Wiuf, Uniform approximation of solutions by elimination of intermediate species in deterministic reaction networks, SIAM J. Appl. Dyn. Syst., 16 (2017), 2259–2286.

10. P. T. Cardin and M. A. Texeira, Fenichel theory for multiple time scale singular perturbation problems, SIAM J. Appl. Dyn. Syst., 16 (2017), 1425–1452.

11. A. Goeke and S. Walcher, A constructive approach to quasi-steady state reduction, J. Math. Chem., 52 (2014), 2596–2626.

12. A. Goeke, S. Walcher and E. Zerz, Determining "small parameters" for quasi-steady state, J. Differ. Equations, 259 (2015), 1149–1180.

13. L. Noethen and S. Walcher, Tikhonov's theorem and quasi-steady state, Discrete Contin. Dyn. Syst. Ser. B, 16 (2011), 945–961.

14. J. Keener and J. Sneyd, Mathematical physiology I: Cellular physiology, 2nd edition, Springer- Verlag, New York, 2009.

15. A. Goeke, C. Schilli, S. Walcher, et al., Computing quasi-steady state reductions, J. Math. Chem., 50 (2012), 1495–1513.

16. A. Goeke, S. Walcher and E. Zerz, Classical quasi-steady state reduction – A mathematical characterization, Physica D, 345 (2017), 11–26.

17. F. R. Gantmacher, Applications of the theory of matrices, Dover, Mineola, 2005.

18. A. Goeke, Reduktion und asymptotische Reduktion von Reaktionsgleichungen., Ph.D thesis, RWTH Aachen, 2013. Available from: http://darwin.bth.rwth-aachen.de/opus3/ volltexte/2013/4814/pdf/4814.pdf.

19. R. Heinrich and M. Schauer, Quasi-steady-state approximation in the mathematical modeling of biochemical networks, Math. Biosci., 65 (1983), 155–170.

20. C. Lax and S. Walcher, Singular perturbations and scaling, Discrete Contin. Dyn. Syst. Ser. B, to appear. arXiv:1807.03107

21. J. D. Murray, Mathematical biology. I. An introduction, 3rd edition, Springer, New York, 2002.

22. M. Stiefenhofer, Quasi-steady-state approximation for chemical reaction networks, J. Math. Biol., 36 (1998), 593–609.

23. F. Verhulst, Methods and applications of singular perturbations. Boundary layers and multiple time scale dynamics, Springer-Verlag, New York, 2005.

© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)

Download full text in PDF

Export Citation

Article outline

Show full outline
Copyright © AIMS Press All Rights Reserved