What Is a Bifurcation Diagram and How Do You Create One?

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    Bifurcation Diagram
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SUMMARY

A bifurcation diagram is a graphical representation of the different states of a dynamical system as a parameter is varied, commonly associated with chaos theory. It is one of the three key icons of chaos theory, alongside the Mandelbrot Set and the Lorenz attractor. The discussion highlights the logistics map equation, x_{n+1}=kx_n(1-x_n), as a basis for creating a bifurcation diagram. Tools such as C++ and Mathematica are mentioned for generating detailed diagrams.

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  • Understanding of chaos theory concepts
  • Familiarity with the logistics map equation
  • Basic knowledge of differential equations
  • Experience with programming in C++ or using Mathematica
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  • Research how to create bifurcation diagrams using Mathematica
  • Learn about the logistics map and its implications in chaos theory
  • Explore the relationship between bifurcation diagrams and differential equations
  • Study the Mandelbrot Set and Lorenz attractor for a comprehensive understanding of chaos theory
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Students preparing for exams in chaos theory, mathematicians interested in dynamical systems, and programmers looking to visualize complex systems using tools like C++ and Mathematica.

Zurtex
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Could some explain or point me to somewhere explains what a Bifurcation Diagram is and how to draw one given a non-linear function. Got an exam today and nobody knows what one is. If it has already been 2 hours after I've posted this don't worry, I'll be heading off to the exam hehe. Thanks if anyone can give any help.
 
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Too late now, thanks if anyone tried to look something up.
 
Zurtex said:
Too late now, thanks if anyone tried to look something up.

Jesus Zurtex. I'm a day late and a dollar short, alright 5 whatever. Wish I would have seen this but I was away for awhile. Has a bifurcation diagram ever been presented in the forum? You know it's one of the three icons of Chaos Theory right? You know, Mandelbrot Set, Lorenz attractor, Feigenbaum plot (bifurcation diagram). We should post one. Well me if you don't feel like it. In fact, since we have three spaces down there, we should post all three. Dude, you're long gone with this aren't you.
 
Attached is the bifurcation diagram for the logistics map. You can make a much more detailed one using C++ but I can't convert it to JPEG so I have to use Mathematica.

x_{n+1}=kx_n(1-x_n)

I suppose it is relevant to differential equations since bifurcation diagrams can also be created for differential equations (pretty sure anyway, need to check). Zurtex, wanna' generate one for such? I suspect I know what you're thinking, "dude, I ain't generating nothin', it's summer time". :smile:
 

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