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The Logistic Map in Action - Transience Divine
January 23rd, 2014
11:43 pm

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The Logistic Map in Action
Everyone's heard of the logistic map:
x_{n+1} = r x_n (1 - x_n)

It's elegant, it's powerful (a classic for modeling ecosystems, e.g.), and it's incredibly chaotic. As you change r, its internal frequency doubles, and then redoubles over a shorter span, and then again and again, until it reaches an infinity frequency over a finite distance. So you get beautiful fractal pictures like the following, bursting with internal structure:
Static Logistic Map

But they rarely tell you how you get the picture, or what it means. The closest you get is that these are "asymptotic" values-- a meaningless statement for something that never settles down.

So, I made an animation. In it, I just keep adding new points, each with a value of r and an initial value of x, and let them fly.

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