Someone asked me what this beautiful diagram meant. Here is the question,
and how I explained it:
Steve Robinson wrote:
Interesting web site. I liked the fractal image, but was confused
by the statement:
" It shows what happens when the output from the function a=k*b*(1-b) gets
fed into itself, as k varies. "
I understand K varying, but I am not much up on Chaos Theory. What
does it mean to say that the output from the function (a), gets fed into
itself? The variable a doesn't show up on the right hand side of the
What is being graphed? X,Y?
Guy Brandenburg replied:
You could think of the b-axis being the horizontal one
(the one that people usually call the x-axis), and the a-axis being the
vertical one (what people call the y-axis most of the time).
So if you re-write it in that manner, you have
y = k*x(1-x).
If you pick a positive value for k, and graph this you
will notice that it is a parabola with its vertex pointing up and its two
arms pointing down. It hits the x axis at (0,0) and (1,0). The value of
the parameter k controls how narrow or wide the 'arms' are.
When I say the output get fed into itself, I mean this.
Suppose we pick k to be 1, to keep things simple. We will restrict x to
values in the interval [0, 1], so we are only looking at the section of
the parabola above the x-axis.
Now pick some random value of x, say 0.3. The equation
is y = x(1-x), so we have y = 0.3*0.7 = 0.21. We now feed that output, that
0.21, back into our equation y=x(1-x). Continue looping the output (y) back
into the next step. You will notice that the values of x and y slowly go
towards zero, where they will remain forever. ( I get 0.21, then 0.1659,
then 0.13837719, and so on, slowly dwindling to zero)
However, now watch what happens when k = 2.5. No matter
what you originally put in for x, the values of y and x soon converge to
But if you try k=3.2, I find that you soon get a double-loop,
so to speak: the values converge to 2 final output values, that alternate:
about 0.513 and 0.799.
If you make k = 4, then things become totally chaotic.
In other words, the behavior of the function f(x) = k*x*(1-x)
when the values of the function are put back into as values of x, depends
on the value of k. (As long as x is restricted to starting in the domain
So the graph can be interpreted thusly:
The horizontal axis is the value of k.
The vertical axis is/are the value(s) of the output(x)
of f(x) after the first 100 terms or so.
So if you pick a point on the bottom edge of the graph,
you have picked a value of k. Read up; you will hit one, two, four, eight,
or nearly uncountably many points. That tells you how many output values
there are for that value of x, AND how big they are.
This function actually has been used by wildlife biologists
to model how fish, plankton, and other critters reproduce themselves. It
was originally thought that if you fix the conditions (k), that this model
would have all the populations (represented as a percent or a decimal fraction
of the theoretical carrying capacity of the environment) reaching some sort
of stable equilibrium. But, no - they found that for various values of k,
the populations will oscillate wildly from generation to generation. In
fact, well before I read about this, I read an article in Scientific American
(over 20 eyars ago) about populations of snowshoe rabbits and lynxes on
some island in Canada that were forever 'balooning' and then crashing, without
anybody coming in from the outside and doing anything to upset anything.
I hope that helps.
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