Image:LogisticMap BifurcationDiagram.png

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Image:LogisticMap BifurcationDiagram.png

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Summary

A bifurcation diagram for the Logistic map:

$x_{n+1}=rx_n(1-x_n)\,$

The horizontal axis is the r parameter, the vertical axis is the x variable. The image was created by forming a 1601 x 1001 array representing increments of 0.001 in r and x. A starting value of x=0.25 was used, and the map was iterated 1000 times in order to stabilize the values of x. 100,000 x -values were then calculated for each value of r and for each x value, the corresponding (x,r) pixel in the image was incremented by one. All values in a column (corresponding to a particular value of r) were then multiplied by the number of non-zero pixels in that column, in order to even out the intensities. Values above 250,000 were set to 250,000, and then the entire image was normalized to 0-255. Finally, pixels for values of r below 3.57 were darkened to increase visibility.

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File history

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	Date/Time	Dimensions	User	Comment
current	00:11, 14 September 2005	1,838×1,300 (571 KB)	PAR	( Bifurcation diagram for the logistic map for parameter r=2.4 to 4.0.)

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Chaos theory

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