bifurcation
Someone asked me what this beautiful diagram meant. Here is the question, and how I explained it:

Steve Robinson wrote:

Hi Guy,
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 equation.

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 0.6.

    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 [0, 1].
   
    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.

Guy

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