In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the x-axis of the plot and the y-axis shows the possible long-term population values of the logistic function. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs convergent series to the Feigenbaum constant. An interesting feature of this diagram is that as the periods go to infinity, r remains finite. When r is greater than (approximately) 3.57, the orbits become chaotic. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple non-linear systems. Chaos theory
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. --User:Lenthe 16:29, 17 Jun 2005 (UTC) ---- :''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. User: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. -User:PierreAbbat 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. User:CyborgTosser (User_talk:CyborgTosser) 09:05, 26 Apr 2005 (UTC) Here it is. [http://abacus.bates.edu/~sross/bifurandorbit01.html] User:CyborgTosser (User_talk:CyborgTosser) 01:47, 27 Apr 2005 (UTC)
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Bifurcation diagram
In mathematics, particularly in dynamical systems, a bifurcation diagram shows the possible long-term values a variable of a system can obtain in function of a parameter of the system. An example is the bifurcation diagram of the logistic map. In this case, the parameter r is shown on the x-axis of the plot and the y-axis shows the possible long-term population values of the logistic function. The bifurcation diagram nicely shows the forking of the possible periods of stable orbits from 1 to 2 to 4 to 8 etc. The ratio of the lengths of successive intervals between values of r for which bifurcation occurs convergent series to the Feigenbaum constant. An interesting feature of this diagram is that as the periods go to infinity, r remains finite. When r is greater than (approximately) 3.57, the orbits become chaotic. Hence this bifurcation diagram demonstrates a nice example of the importance of chaos theory in even very simple non-linear systems. Chaos theory
Bifurcation diagram
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. --User:Lenthe 16:29, 17 Jun 2005 (UTC) ---- :''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. User: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. -User:PierreAbbat 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. User:CyborgTosser (User_talk:CyborgTosser) 09:05, 26 Apr 2005 (UTC) Here it is. [http://abacus.bates.edu/~sross/bifurandorbit01.html] User:CyborgTosser (User_talk:CyborgTosser) 01:47, 27 Apr 2005 (UTC)
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