Imagine some sort of a polynomial function f(x) = r * x * (1-x) that is also depending on some constant r in range 0 < r < 1.
You apply x = f(x) for many times, at your wish. Think of it as a for-loop, or f(f(f(…f(f(x))…))). The resulting value will reside (converge) to some value.
So, this looped function sets to some stable point with any r in range until x < 3. After that, it doesn’t converge but it will constantly jump between one of 2 points, then 4, 8, … and at some point it goes wacky-macky.
The graph from Wikipedia merely shows the probability of the value f(f(f(...f(f(x))...))) for all r in the given range.
ELI5:
Imagine some sort of a polynomial function
f(x) = r * x * (1-x)that is also depending on some constantrin range0 < r < 1.You apply
x = f(x)for many times, at your wish. Think of it as a for-loop, or f(f(f(…f(f(x))…))). The resulting value will reside (converge) to some value.So, this looped function sets to some stable point with any
rin range untilx < 3. After that, it doesn’t converge but it will constantly jump between one of 2 points, then 4, 8, … and at some point it goes wacky-macky.The graph from Wikipedia merely shows the probability of the value
f(f(f(...f(f(x))...)))for allrin the given range.If you’re curious, Veritasium’s made a good explanation here: https://youtu.be/ovJcsL7vyrk
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