Contraposition: Difference between revisions
→Simple proof using Venn diagrams: fix "blonde US girls" example to be consistent wrt the thing we want to prove |
en>HLwiKi →Simple proof using Euler diagrams: Rename section to "Intuitive explanation"; improve the text a little. |
||
| Line 1: | Line 1: | ||
[[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).]] | |||
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. | |||
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). | |||
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. | |||
The [[normal form]]{{dn|date=December 2013}} of the Bogdanov–Takens bifurcation is | |||
:<math> \begin{align} | |||
y_1' &= y_2, \\ | |||
y_2' &= \beta_1 + \beta_2 y_1 + y_1^2 \pm y_1 y_2. | |||
\end{align} </math> | |||
It has also been found the existence of a codimension-three degenerate Takens–Bogdanov bifurcation, also known as [[Dumortier–Roussarie–Sotomayor]] bifurcation. | |||
==References== | |||
*Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373–388, 1981. | |||
*Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995. | |||
*Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2, 1–111, 1974. | |||
*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). | |||
==External links== | |||
* {{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}} | |||
{{DEFAULTSORT:Bogdanov-Takens bifurcation}} | |||
[[Category:Bifurcation theory]] | |||
Revision as of 03:39, 21 November 2013
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.
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).
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.
The normal formTemplate:Dn of the Bogdanov–Takens bifurcation is
It has also been found the existence of a codimension-three degenerate Takens–Bogdanov bifurcation, also known as Dumortier–Roussarie–Sotomayor bifurcation.
References
- Bogdanov, R. "Bifurcations of a Limit Cycle for a Family of Vector Fields on the Plane." Selecta Math. Soviet 1, 373–388, 1981.
- Kuznetsov, Y. A. Elements of Applied Bifurcation Theory. New York: Springer-Verlag, 1995.
- Takens, F. "Forced Oscillations and Bifurcations." Comm. Math. Inst. Rijksuniv. Utrecht 2, 1–111, 1974.
- 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).