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The Bifurcation Diagram

Plotting vertically the stable values of the logistic map — that is, the attractors — against the values of the coefficient a yields a very complex fractal.

Brief Explanation

Here is yet another way to visualize the complex dynamics of the logistic map. For each value of a, we iterate the function until it “settles down” and then plot just those values. Of course, if we are in the chaotic region, things never really settle, so we (arbitrarily) just choose a number of points (here, 40) to plot.

If you are unfamiliar with the logistic map, you should visit the preceding pages in this series, by clicking the “Previous” link at the top of this page.

To use this page, just drag the a-slider back and forth. As the value of a changes, the graph shows in the vertical axis what are the stable values of the logistic map a∙x∙(1-x). You can change the range of a by entering new values in the text fields on either side of the slider. This helps you zoom in on particular regions. The list of buttons below also lets you automatically set the regions to particularly interesting ones.



Try It Yourself


Minimum a Slider for setting a Maximum a

Interesting values for a



Remarks

This diagram contains many copies of itself — stretched and bent somewhat, but still in there! The simplest example comes from just looking at the first period-doubling region. Each stable value bifurcates into a pair just like the other stable values do. If we plotted just the values from each “strand” we would get a graph that looks a lot like the whole one. Just warped a bit. This goes on infinitely! Quite mind-boggling.

And yet, out of the confusion there sometimes arise simple attractors of low period. In fact, there is a theorem that shows that if any iterated function has a period-3 cycle in it (which this logistic map certainly does) the function will have periodic cycles for all integer values. Somewhere. Finding them might be hard. But it is still quite amazing.



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