Export file:


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


  • 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.
  Article Metrics

Keywords reaction network; dimension reduction; invariant set; multiple time scales

Citation: Niclas Kruff, Sebastian Walcher. Coordinate-independent singular perturbation reduction for systems with three time scales. Mathematical Biosciences and Engineering, 2019, 16(5): 5062-5091. doi: 10.3934/mbe.2019255


  • 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.


This article has been cited by

  • 1. Ian Lizarraga, Martin Wechselberger, Computational Singular Perturbation Method for Nonstandard Slow-Fast Systems, SIAM Journal on Applied Dynamical Systems, 2020, 19, 2, 994, 10.1137/19M1242677

Reader Comments

your name: *   your email: *  

© 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

Copyright © AIMS Press All Rights Reserved