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 attractor of the system. Typical local examples are the Hopf...
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsReferences
Danino T, Mondragn-Palomino O, Tsimring L, Hasty J (2010) A synchronized quorum of genetic clocks. Nature 463(7279):326–330
Fall CP, Marland ES, Wagner JM, Tyson JJ (2002) Computational cell biology. Springer, New York, In memory of Joel Keizer
Gardner TS, Cantor CR, Collins JJ (2000) Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767):339–342
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 277–308
Hoyle R (2006) Pattern formation: an introduction to methods. Cambridge University Press, Cambridge
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):869–876
Krauskopf B, Osinga HM, Galn-Vioque J (2007) Numerical continuation methods for dynamical systems: path following and boundary value problems. Springer, New York
Kuznetsov YA (2004) Elements of applied bifurcation theory, 3rd edn. Springer, New York
Murray JD (2007) Mathematical biology. Springer, New York, third (in two parts) edition
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 5:332
Shilnikov LP, Shilnikov AL, Turaev DV, Chua LO (2001) Methods of qualitative theory in nonlinear dynamics. World Scientific, Singapore
Sieber J, Gonzalez-Buelga A, Neild SA, Wagg DJ, Krauskopf B (2008) Experimental continuation of periodic orbits through a fold. Phys Rev Lett 100:244101
Sneed J, Keener J (2008) Mathematical physiology. Springer, New York, second, in two parts edition
Strogatz SH (1994) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. Westview, Boulder
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media, LLC
About this entry
Cite this entry
Champneys, A., Tsaneva-Atanasova, K. (2013). Dynamical Systems Theory, Bifurcation Analysis. In: Dubitzky, W., Wolkenhauer, O., Cho, KH., Yokota, H. (eds) Encyclopedia of Systems Biology. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9863-7_270
Download citation
DOI: https://doi.org/10.1007/978-1-4419-9863-7_270
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-9862-0
Online ISBN: 978-1-4419-9863-7
eBook Packages: Biomedical and Life SciencesReference Module Biomedical and Life Sciences