Encyclopedia of Systems Biology

2013 Edition
| Editors: Werner Dubitzky, Olaf Wolkenhauer, Kwang-Hyun Cho, Hiroki Yokota

Dynamical Systems Theory, Bifurcation Analysis

  • Alan ChampneysEmail author
  • Krasimira Tsaneva-Atanasova
Reference work entry
DOI: https://doi.org/10.1007/978-1-4419-9863-7_270

Synonyms

Definition

Bifurcation theory refers to the study of qualitative changes to the state of a system as a parameter is varied. It can be applied to  steady state systems, or to dynamical systems and can be understood best at the level of a mathematical model, although recent techniques allow the method to be applied to experiments with feedback control. Typically the theory is applied to a  continuous model, but can also be used in  discrete models and mathematics, difference equations. There are dedicated numerical implementations of bifurcation theory using path-following, or numerical continuation. There is a distinction between a local  bifurcation, which can be understood in terms of a change to the number or stability of simple steady states, and a global bifurcation, which cannot. Often global bifurcations cause catastrophic changes to the  attractorof the system. Typical local examples...

This is a preview of subscription content, log in to check access.

References

  1. Danino T, Mondragn-Palomino O, Tsimring L, Hasty J (2010) A synchronized quorum of genetic clocks. Nature 463(7279):326–330PubMedCrossRefGoogle Scholar
  2. Fall CP, Marland ES, Wagner JM, Tyson JJ (2002) Computational cell biology. Springer, New York, In memory of Joel KeizerGoogle Scholar
  3. Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767):339–342PubMedCrossRefGoogle Scholar
  4. Golubitsky M, Josic K, Kaper TJ (2001) An unfolding theory approach to bursting in fast-slow systems. Global Analysis of Dynamical Systems. Festschrift dedicated to Floris Takens for his 60th birthday, pp 277–308Google Scholar
  5. Hoyle R (2006) Pattern formation: an introduction to methods. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  6. Huang S, Ernberg I, Kauffman S (2009) Cancer attractors: a systems view of tumors from a gene network dynamics and developmental perspective. Semin Cell Dev Biol 20(7):869–876PubMedCrossRefGoogle Scholar
  7. Krauskopf B, Osinga HM, Galn-Vioque J (2007) Numerical continuation methods for dynamical systems: path following and boundary value problems. Springer, New YorkCrossRefGoogle Scholar
  8. Kuznetsov YA (2004) Elements of applied bifurcation theory, 3rd edn. Springer, New YorkCrossRefGoogle Scholar
  9. Murray JD (2007) Mathematical biology. Springer, New York, third (in two parts) editionGoogle Scholar
  10. Shankaran H, Ippolito DL, Chrisler WB, Resat H, Bollinger N, Opresko LK, Wiley HS (2009) Rapid and sustained nuclear-cytoplasmic erk oscillations induced by epidermal growth factor. Mol Syst Biol 5:332PubMedCrossRefGoogle Scholar
  11. Shilnikov LP, Shilnikov AL, Turaev DV, Chua LO (2001) Methods of qualitative theory in nonlinear dynamics. World Scientific, SingaporeGoogle Scholar
  12. Sieber J, Gonzalez-Buelga A, Neild SA, Wagg DJ, Krauskopf B (2008) Experimental continuation of periodic orbits through a fold. Phys Rev Lett 100:244101PubMedCrossRefGoogle Scholar
  13. Sneed J, Keener J (2008) Mathematical physiology. Springer, New York, second, in two parts editionGoogle Scholar
  14. Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview, BoulderGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Department of Engineering MathematicsUniversity of BristolBristolUK