Parametric Qualitative Analysis of Ordinary Differential Equations: Computer Algebra Methods for Excluding Oscillations (Extended Abstract) (Invited Talk)

  • Andreas Weber
  • Thomas Sturm
  • Werner M. Seiler
  • Essam O. Abdel-Rahman
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6244)


Investigating oscillations for parametric ordinary differential equations (ODEs) has many applications in science and engineering but is a very hard problem. Already for two dimensional polynomial systems this question is related to Hilbert’s 16th problem, which is still unsolved [1].

Using the theory of Hopf-bifurcations some non-numeric algorithmic methods have been recently developed to determine ranges of parameters for which some small stable limit cycle will occur in the system [2,3,4,5,6,7,8]. These algorithms give exact conditions for the existence of fixed points undergoing a Poincar’e Andronov-Hopf bifurcation that give birth to a small stable limit cycle under some general conditions which can be made algorithmic, too. If these conditions are not satisfied, one can be sure that there are no such fixed points, but unfortunately one cannot conclude that there are no limit cycles–which could arise by other means. Nevertheless, it is tempting to conjecture even in these cases that there are no oscillations, as has been done e.g. in [5,6].


Symbolic Computation Polynomial Vector Genetic Circuit Cylindrical Algebraic Decomposition Dulac Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Ilyashenko, Y.: Centennial history of Hilbert’s 16th Problem. Bull. Am. Math. Soc., New Ser. 39(3), 301–354 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    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, Special issue on ‘Symbolic Computation in Biology’ 2(3), 493–515 (2009)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Niu, W., Wang, D.: Algebraic approaches to stability analysis of biological systems. Mathematics in Computer Science 1(3), 507–539 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Sturm, T., Weber, A.: Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In: Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H. (eds.) AB 2008. LNCS, vol. 5147, pp. 200–215. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  5. 5.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P., Ürgüplü, A.: On proving the absence of oscillations in models of genetic circuits. In: Anai, H., Horimoto, K., Kutsia, T. (eds.) Ab 2007. LNCS, vol. 4545, pp. 66–80. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Boulier, F., Lefranc, M., Lemaire, F., Morant, P.E.: Applying a rigorous quasi-steady state approximation method for proving the absence of oscillations in models of genetic circuits. In: Horimoto, K., Regensburger, G., Rosenkranz, M., Yoshida, H. (eds.) AB 2008. LNCS, vol. 5147, pp. 56–64. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    El Kahoui, M., Weber, A.: Symbolic equilibrium point analysis in parameterized polynomial vector fields. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) Computer Algebra in Scientific Computing (CASC 2002), Yalta, Ukraine, pp. 71–83 (September 2002)Google Scholar
  8. 8.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hong, H., Liska, R., Steinberg, S.: Testing stability by quantifier elimination. Journal of Symbolic Computation 24(2), 161–187 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bendixson, I.: Sur les curbes définiés par des équations différentielles. Acta Math. 24, 1–88 (1901)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guckenheimer, J., Holmes, P.: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Applied Mathematical Sciences, vol. 42. Springer, Heidelberg (1983)CrossRefzbMATHGoogle Scholar
  13. 13.
    Dulac, H.: Recherche des cycles limites. CR Acad. Sci. Paris 204, 1703–1706 (1937)zbMATHGoogle Scholar
  14. 14.
    Weispfenning, V.: The complexity of linear problems in fields. Journal of Symbolic Computation 5(1&2), 3–27 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. Journal of Symbolic Computation 5(1-2), 29–35 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Strzebonski, A.: Solving systems of strict polynomial inequalities. Journal of Symbolic Computation 29(3), 471–480 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Strzebonski, A.W.: Cylindrical algebraic decomposition using validated numerics. J. Symb. Comput. 41(9), 1021–1038 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Brown, C.W.: QEPCAD B: A system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bulletin 38(1), 23–24 (2004)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Dolzmann, A., Sturm, T.: REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin 31(2), 2–9 (1997)CrossRefGoogle Scholar
  20. 20.
    Sturm, T.: Redlog online resources for applied quantifier elimination. Acta Academiae Aboensis, Ser. B 67(2), 177–191 (2007)Google Scholar
  21. 21.
    Weispfenning, V.: Quantifier elimination for real algebra—the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing 8(2), 85–101 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Dolzmann, A., Sturm, T.: Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation 24(2), 209–231 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Weispfenning, V.: A new approach to quantifier elimination for real algebra. In: Caviness, B., Johnson, J. (eds.) Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and Monographs in Symbolic Computation, pp. 376–392. Springer, Wien (1998)CrossRefGoogle Scholar
  24. 24.
    Gilch, L.A.: Effiziente Hermitesche Quantorenelimination. Diploma thesis, Universität Passau, D-94030 Passau, Germany (September 2003)Google Scholar
  25. 25.
    Dolzmann, A., Gilch, L.A.: Generic Hermitian quantifier elimination. In: Buchberger, B., Campbell, J. (eds.) AISC 2004. LNCS (LNAI), vol. 3249, pp. 80–93. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  26. 26.
    Sturm, T.: New domains for applied quantifier elimination. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2006. LNCS, vol. 4194, pp. 295–301. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  27. 27.
    Lasaruk, A., Sturm, T.: Weak quantifier elimination for the full linear theory of the integers. A uniform generalization of Presburger arithmetic. Applicable Algebra in Engineering, Communication and Computing 18(6), 545–574 (2007)CrossRefzbMATHGoogle Scholar
  28. 28.
    Lasaruk, A., Sturm, T.: Weak integer quantifier elimination beyond the linear case. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E.V. (eds.) CASC 2007. LNCS, vol. 4770, pp. 275–294. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  29. 29.
    Weispfenning, V.: Simulation and optimization by quantifier elimination. Journal of Symbolic Computation, Special issue on applications of quantifier elimination 24(2), 189–208 (1997)MathSciNetzbMATHGoogle Scholar
  30. 30.
    Muldowney, J.S.: Compound matrices and ordinary differential equations. Rocky Mt. J. Math. 20(4), 857–872 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Weber, A., Sturm, T., Abdel-Rahman, E.O.: Algorithmic global criteria for excluding oscillations. Bulletin of Mathematical Biology (2010); Accepted for publication. Special issue on “Algebraic Biology”Google Scholar
  32. 32.
    Goriely, A.: Integrability and nonintegrability of dynamical systems. World Scientific, Singapore (2001)CrossRefzbMATHGoogle Scholar
  33. 33.
    Tóth, J.: Bendixson-type theorems with applications. Z. Angew. Math. Mech. 67, 31–35 (1987)CrossRefzbMATHGoogle Scholar
  34. 34.
    Hars, V., Tóth, J.: On the inverse problem of reaction kinetics. In: Farkas, M. (ed.) Colloquia Mathematica Societatis Janos Bolyai, Qualitative Theory of Differential Equations, Szeged, Hungary, pp. 363–379 (1981)Google Scholar
  35. 35.
    Tuckwell, H.C., Wan, F.Y.M.: On the behavior of solutions in viral dynamical models. BioSystems 73(3), 157–161 (2004)CrossRefGoogle Scholar
  36. 36.
    Bonhoeffer, S., Coffin, J.M., Nowak, M.A.: Human immunodeficiency virus drug therapy and virus load. The Journal of Virology 71(4), 3275 (1997)Google Scholar
  37. 37.
    Feckan, M.: A generalization of Bendixson’s criterion. Proceedings American Mathematical Society 129(11), 3395–3400 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    She, Z., Xia, B., Xiao, R., Zheng, Z.: A semi-algebraic approach for asymptotic stability analysis. Nonlinear Analysis: Hybrid Systems 3(4), 588–596 (2009)MathSciNetzbMATHGoogle Scholar
  39. 39.
    Rabier, P.J., Rheinboldt, W.C.: Theoretical and numerical analysis of differential-algebraic equations. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis, vol. VIII, pp. 183–540. North-Holland, Amsterdam (2002)Google Scholar
  40. 40.
    Riaza, R.: Differential-Algebraic Systems. World Scientific, Hackensack (2008)CrossRefzbMATHGoogle Scholar
  41. 41.
    Seiler, W.M.: Involution — The Formal Theory of Differential Equations and its Applications in Computer Algebra. In: Algorithms and Computation in Mathematics, vol. 24. Springer, Berlin (2009)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Andreas Weber
    • 1
  • Thomas Sturm
    • 2
  • Werner M. Seiler
    • 3
  • Essam O. Abdel-Rahman
    • 1
  1. 1.Institut für Informatik IIUniversität BonnBonnGermany
  2. 2.Departamento de Matemáticas, Estadística y Computación, Facultad de CienciasUniversidad de CantabriaSantanderSpain
  3. 3.Institut für MathematikUniversität KasselKasselGermany

Personalised recommendations