Investigating algebraic and logical algorithms to solve Hopf bifurcation problems in algebraic biology

Mathematics in Computer Science

Volumn 2, Issue 3, 2009, Pages 493-515

  Sturm, Thomas ^a   Weber, Andreas ^b   Abdel Rahman, Essam O ^b   El Kahoui, M'hammed ^c  

^a UNIVERSITY OF CANTABRIA   (Spain)

^b UNIVERSITY OF BONN   (Germany)

^c FACULTÉ DES SCIENCES SEMLALIA   (Morocco)

Author keywords

Dynamical systems; Hopf bifurcations; Logical inconsistency checking; Oscillations; Quantifier elimination over the reals

Indexed keywords

EID: 76349103515     PISSN: 16618270     EISSN: 16618289     Source Type: Journal    
DOI: 10.1007/s11786-008-0067-1     Document Type: Article

References (22)

1. F. Boulier, M. Lefranc, F. Lemaire, P. Morant, and A. Ürgüplü. On proving the absence of oscillations in models of genetic circuits. In H. Anai, H. Horimoto, and T. Kutsia, editors, Algebraic Biology (AB 2007), volume 4545 of Lecture Notes in Computer Science, pages 66-80. Springer-Verlag, 2007.
2. F. Boulier, M. Lefranc, F. Lemaire, and P.-E. Morant. Applying a rigorous quasisteady state approximation method for proving the absence of oscillations in models of genetic circuits. In K. Horimoto, G. Regensburger, M. Rosenkranz, and H. Yoshida, editors, Algebraic Biology (AB 2008) - Third International Conference, volume 5147 of Lecture Notes in Computer Science, pages 56-64, Castle of Hagenberg, Austria, August 2008. Springer-Verlag.
3. C. W. Brown. QEPCAD B: A system for computing with semi-algebraic sets via cylindrical algebraic decomposition. ACM SIGSAM Bulletin, 38(1):23-24, 2004.
4. C. W. Brown and C. Groß. Efficient preprocessing methods for quantifier elimination. In V. G. Ganzha, Ernst W. Mayr, and Evgenii V. Vorozhtsov, editors, Computer Algebra in Scientific Computing (CASC'06), volume 4194 of Lecture Notes in Computer Science, pages 89-100, Chisinau, Moldova, September 2006. Springer-Verlag.
5. G. E. Collins and H. Hong. Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation, 12(3):299-328, September 1991.
6. A. Dolzmann and T. Sturm. REDLOG: Computer algebra meets computer logic. ACM SIGSAM Bulletin, 31(2):2-9, June 1997.
7. A. Dolzmann and T. Sturm. Simplification of quantifier-free formulae over ordered fields. Journal of Symbolic Computation, 24(2):209-231, August 1997.
8. A. Dolzmann, T. Sturm, and V. Weispfenning. A new approach for automatic theorem proving in real geometry. Journal of Automated Reasoning, 21(3):357-380, 1998.
9. M. El Kahoui and A. Weber. Deciding Hopf bifurcations by quantifier elimination in a software-component architecture. Journal of Symbolic Computation, 30(2):161-179, August 2000.
10. G. F. Fussmann, S. P. Ellner, K. W. Shertzer, and N. G. Hairston, Jr. Crossing the Hopf bifurcation in a live predator-prey system. Science, 290(5495):1358-1360, 2000.
11. K. Gatermann, M. Eiswirth, and A. Sensse. Toric ideals and graph theory to analyze Hopf bifurcations in mass action systems. Journal of Symbolic Computation, 40(6):1361-1382, 2005.
12. M. Mincheva and M. R. Roussel. Graph-theoretic methods for the analysis of chemical and biochemical networks. I. multistability and oscillations in ordinary differential equation models. Journal of Mathematical Biology, 55(1):61-86, 2007.
13. W. Niu and D. Wang. Algebraic approaches to stability analysis of biological systems. Mathematics in Computer Science, 1(3):507-539, 2008.
14. B. Novak, Z. Pataki, A. Ciliberto, and J. J. Tyson. Mathematical model of the cell division cycle of fission yeast. Chaos: An Interdisciplinary Journal of Nonlinear Science, 11(1):277-286, 2001.
15. A. Sensse, M. J. B. Hauser, and M. Eiswirth. Feedback loops for Shilnikov chaos: The peroxidase-oxidase reaction. The Journal of Chemical Physics, 125:014901-1-12,2006.
16. T. Sturm. Redlog online resources for applied quantifier elimination. Acta Academiae Aboensis, Ser. B, 67(2):177-191, 2007.
17. T. Sturm and A. Weber. Investigating generic methods to solve Hopf bifurcation problems in algebraic biology. In K. Horimoto, G. Regensburger, M. Rosenkranz, and H. Yoshida, editors, Algebraic Biology - Third International Conference (AB 2008), volume 5147 of Lecture Notes in Computer Science, Castle of Hagenberg, Austria, 2008. Springer-Verlag.
18. M. Sun, L. Tian, and J. Yin. Hopf bifurcation analysis of the energy resource chaotic system. International Journal of Nonlinear Science, 1(1):49-53, 2006.
19. J. J. Tyson, K. Chen, and B. Novak. Network Dynamics and Cell Physiology. Nat Rev Mol Cell Biol, 2(12):908-916, 2001.
20. A. Weber. Quantifier elimination on real closed fields and differential equations. In B. Löwe, editor, Algebra, Logic, Set Theory - Festschrift für Ulrich Felgner zum 65. Geburtstag, volume 4 of Studies in Logic, pages 291-315. College Publications, 2007.
21. V. Weispfenning. Quantifier elimination for real algebra - the quadratic case and beyond. Applicable Algebra in Engineering Communication and Computing, 8(2):85-101, February 1997.
22. V. Weispfenning. Simulation and optimization by quantifier elimination. Journal of Symbolic Computation, 24(2):189-208, August 1997. Special issue on applications of quantifier elimination.