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Efficient Methods to Compute Hopf Bifurcations in Chemical Reaction Networks Using Reaction Coordinates

  • Hassan Errami
  • Markus Eiswirth
  • Dima Grigoriev
  • Werner M. Seiler
  • Thomas Sturm
  • Andreas Weber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8136)

Abstract

We build on our previous work to compute Hopf bifurcation fixed points for chemical reaction systems on the basis of reaction coordinates. For determining the existence of Hopf bifurcations the main algorithmic problem is to determine whether a single multivariate polynomial has a zero for positive coordinates. For this purpose we provide heuristics on the basis of the Newton polytope that ensure the existence of positive and negative values of the polynomial for positive coordinates. We apply our method to the example of the Methylene Blue Oscillator (MBO).

Keywords

Hopf Bifurcation Lorenz System Hopf Bifurcation Point Toric Ideal Newton Polytope 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Errami, H., Seiler, W.M., Eiswirth, M., Weber, A.: Computing hopf bifurcations in chemical reaction networks using reaction coordinates. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2012. LNCS, vol. 7442, pp. 84–97. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  2. 2.
    El Kahoui, M., Weber, A.: Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. Journal of Symbolic Computation 30(2), 161–179 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Sturm, T., Weber, A., Abdel-Rahman, E.O., El Kahoui, M.: Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology. Mathematics in Computer Science 2(3), 493–515 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Clarke, B.L.: Stability of Complex Reaction Networks. Advances in Chemical Physics, vol. XLIII. Wiley Online Library (1980)Google Scholar
  5. 5.
    Gatermann, K., Eiswirth, M., Sensse, A.: Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. Journal of Symbolic Computation 40(6), 1361–1382 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer (1990)Google Scholar
  7. 7.
    Orlando, L.: Sul problema di hurwitz relativo alle parti reali delle radici di un’equazione algebrica. Mathematische Annalen 71(2), 233–245 (1911)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Gantmacher, F.R.: Application of the Theory of Matrices. Interscience Publishers, New York (1959)Google Scholar
  9. 9.
    Porter, B.: Stability Criteria for Linear Dynamical Systems. Academic Press, New York (1967)Google Scholar
  10. 10.
    Yu, P.: Closed-form conditions of bifurcation points for general differential equations. International Journal of Bifurcation and Chaos 15(4), 1467–1483 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Liu, W.M.: Criterion of Hopf bifurcations without using eigenvalues. Journal of Mathematical Analysis and Applications 182(1), 250–256 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dolzmann, A., Sturm, T.: Redlog: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8(2), 85–101 (1997)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Dolzmann, A., Sturm, T., Weispfenning, V.: Real quantifier elimination in practice. In: Matzat, B.H., Greuel, G.M., Hiss, G. (eds.) Algorithmic Algebra and Number Theory, pp. 221–247. Springer, Heidelberg (1998)Google Scholar
  15. 15.
    Dolzmann, A., Seidl, A., Sturm, T.: Efficient projection orders for CAD. In: Gutierrez, J. (ed.) Proceedings of the 2004 International Symposium on Symbolic and Algebraic Computation (ISSAC 2004), pp. 111–118. ACM Press, New York (2004)CrossRefGoogle Scholar
  16. 16.
    Lorenz, E.N.: Deterministic nonperiodic flow. Journal of the Atmospheric Sciences 20(2), 130–141 (1963)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Rand, R.H., Armbruster, D.: Perturbation Methods, Bifurcation Theory and Computer Algebra. Applied Mathematical Sciences, vol. 65. Springer (1987)Google Scholar
  18. 18.
    Brown, C.W., El Kahoui, M., Novotni, D., Weber, A.: Algorithmic methods for investigating equilibria in epidemic modeling. Journal of Symbolic Computation 41(11), 1157–1173 (2006)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Sturmfels, B.: Solving Systems of Polynomial Equations. AMS, Providence (2002)zbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Hassan Errami
    • 1
  • Markus Eiswirth
    • 2
    • 3
  • Dima Grigoriev
    • 4
  • Werner M. Seiler
    • 1
  • Thomas Sturm
    • 5
  • Andreas Weber
    • 6
  1. 1.Institut für MathematikUniversität KasselKasselGermany
  2. 2.Fritz-Haber-Institut der Max-Planck-GesellschaftBerlinGermany
  3. 3.Ertl Center for Electrochemisty and CatalysisGISTSouth Korea
  4. 4.CNRS, MathématiquesUniversité de LilleVilleneuve d’AscqFrance
  5. 5.Max-Planck-Institut für InformatikSaarbrückenGermany
  6. 6.Institut für Informatik IIUniversität BonnBonnGermany

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