Talk:Bifurcation diagram
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I have deleted most of this text since it was about the logistic map or about the mandelbrot set, but not about bifurcation diagrams and most of its contents was already contained under logistic map. I am not an expert on dynamical systems, so I hope someone else can write the appropriate article for this title. --Lenthe 16:29, 17 Jun 2005 (UTC)
When Xn+1 = R Xn another good way to show the stable points is to plot Xn+1 as function of Xn, with Excel it is easy to do, it can be verified that the system is chaotic when R is slightly higher than 3,57 (it will become non chaotic for higher values of R)
by Dingy, July 2005, figures homemade, can be used in the text
- An important theorem about bifurcation diagrams is Sarkovskii's theorem, summarized as "Period three implies chaos".
Sarkovskii's theorem is not about bifurcation diagrams, since it doesn't say anything about how the cycle structure changes as the parameter is varied. Also "Period three implies chaos" is not Sarkovskii's theorem, and does not follow from it. AxelBoldt 02:17 Sep 30, 2002 (UTC)
The order in which the cycles first appear, starting with the 1-cycle and ending with the 3-cycle, is the same as Sarkovskii's order backward. "Period Three Implies Chaos" is by James Yorke. I haven't read it, but according to Chaos by James Gleick, when Yorke was in Berlin, Sarkovskii came up to him and told him that he had proved the same result. -phma
The diagram shown here is not a bifurcation diagram. It is an orbit diagram (which is unfortunately sometimes incorrectly referred to as a bifurcation diagram). A bifurcation diagram shows all cycles, attracting or otherwise, and does not include points that have not yet converged to a cycle, since it is not generated by iteration. I know of a good reference that explains the difference and has examples of both. I will try to find it. CyborgTosser (Only half the battle) 09:05, 26 Apr 2005 (UTC)
Here it is. [1] CyborgTosser (Only half the battle) 01:47, 27 Apr 2005 (UTC)
Sarkovskii's theorem says that "period three" implies an infinity of periodic orbits, which is often one of the defining characteristics of chaos. However, this doesn't really have anything to do with bifurcation diagrams. This pages should perhaps contain bifurcation diagrams for the three simple local bifurcations (saddle-node, transcritical and pitchfork) and perhaps also for the Hopf bifurcation. The logistic map diagram is not a complete bifurcation diagram because it does not contain the unstable solutions after each of the period-doubling bifurcations, althought it is often referred to as such. Mathmoclaire 05:14, 13 May 2006 (UTC)
[edit] Rewrite
I'm going to try and attempt to rewrite this page, as currently it's not really about bifurcation diagrams per se. If anyone has any comments or would like to help, that would be great. Mathmoclaire 13:19, 29 July 2006 (UTC)
