Contraposition - Revision history

en>Miracle Pen at 19:38, 3 January 2015

2015-01-03T19:38:52Z

← Older revision		Revision as of 21:38, 3 January 2015
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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)~~.]]		Friends call him Royal Seyler. Arizona has always been my residing location but my wife desires us to move. My job is a messenger. Playing croquet is something I will never give up.<br><br>Also visit my site: [http://www.tenx.com/UserProfile/tabid/62/userId/13166/Default.aspx extended car warranty]

	~~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]~~]

en>HLwiKi: /* Simple proof using Euler diagrams */ Rename section to "Intuitive explanation"; improve the text a little.

2013-11-21T02:39:52Z

Simple proof using Euler diagrams: Rename section to "Intuitive explanation"; improve the text a little.

← Older revision		Revision as of 04:39, 21 November 2013
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	~~Hello~~, ~~my name is Felicidad but I don~~'~~t like when individuals use my full title~~. ~~Delaware~~ is ~~our beginning place~~. ~~What he truly enjoys doing~~ is ~~to perform handball but he is struggling to find time for~~ it. ~~Interviewing~~ is ~~what I do in my working day occupation.~~<br><br>~~Feel free to visit my~~ web ~~blog; [~~http://~~Jvawest~~.~~com/UserProfile/tabid/152/userId~~/~~2157~~/~~Default~~.~~aspx Jvawest.com~~]		[[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]]

80.237.156.64: /* Simple proof using Venn diagrams */ fix "blonde US girls" example to be consistent wrt the thing we want to prove

2012-08-04T21:01:24Z

Simple proof using Venn diagrams: fix "blonde US girls" example to be consistent wrt the thing we want to prove

New page

Hello, my name is Felicidad but I don't like when individuals use my full title. Delaware is our beginning place. What he truly enjoys doing is to perform handball but he is struggling to find time for it. Interviewing is what I do in my working day occupation.<br><br>Feel free to visit my web blog; [http://Jvawest.com/UserProfile/tabid/152/userId/2157/Default.aspx Jvawest.com]