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| [[Image:Bogdanov takens bifurcation.svg|350px|right|thumb|Bifurcation diagrams with parameters ''β''<sub>1</sub>, ''β''<sub>2</sub> = (from top-left to bottom-right): (−1,1), (1/4,−1), (1,0), (0,0), (−6/25,−1), (0,1).]]
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| In [[bifurcation theory]], a field within [[mathematics]], a '''Bogdanov–Takens bifurcation''' is a well-studied example of a bifurcation with [[co-dimension]] two, meaning that two parameters must be varied for the bifurcation to occur. It is named after [[Rifkat Bogdanov]] and [[Floris Takens]], who independently and simultaneously described this bifurcation.
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| A system ''y''' = ''f''(''y'') undergoes a Bogdanov–Takens bifurcation if it has a fixed point and the linearization of ''f'' around that point has a double [[eigenvalue]] at zero (assuming that some technical nondegeneracy conditions are satisfied).
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| Three codimension-one bifurcations occur nearby: a [[saddle-node bifurcation]], an [[Andronov–Hopf bifurcation]] and a [[homoclinic bifurcation]]. All associated bifurcation curves meet at the Bogdanov–Takens bifurcation.
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| The [[normal form]]{{dn|date=December 2013}} of the Bogdanov–Takens bifurcation is
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| :<math> \begin{align}
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| y_1' &= y_2, \\
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| y_2' &= \beta_1 + \beta_2 y_1 + y_1^2 \pm y_1 y_2.
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| \end{align} </math>
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| It has also been found the existence of a codimension-three degenerate Takens–Bogdanov bifurcation, also known as [[Dumortier–Roussarie–Sotomayor]] bifurcation.
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| ==References==
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| *Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373–388, 1981.
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| *Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
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| *Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2, 1–111, 1974.
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| *Dumortier F., Roussarie R., Sotomayor J. and Zoladek H., Bifurcations of Planar Vector Fields, Lecture Notes in Math. vol. 1480, 1–164, Springer-Verlag (1991).
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| ==External links==
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| * {{cite web| title=Bogdanov–Takens Bifurcation| url=http://www.scholarpedia.org/article/Bogdanov-Takens_Bifurcation| last=Guckenheimer| first=John| coauthors=Yuri A. Kuznetsov| year=2007| work=Scholarpedia| accessdate=2007-03-09| authorlink=John Guckenheimer}}
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| {{DEFAULTSORT:Bogdanov-Takens bifurcation}}
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| [[Category:Bifurcation theory]]
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