Yoav Ben-Dov

the logistic equation

a simple demonstration of chaos

what is the logistic equation?

The logistic equation is a simple example of chaos. It was first studied on computer by the theoretical population biologist Robert May in the late 1960's.

It is expressed in the mathematical formula:

This formula takes a number x, which can be any number between 0 and 1, and changes it into a new value. The new value of x can then be fed again into the formula, and so on. This is called an iteration process.

k is a number between 0 and 4, called "a parameter". It is kept constant while the iteration process is run. It turnes out that whatever the initial value of x, after some 10s or 100s of iterations, the values of x settle into a definite pattern. This pattern can be constant, periodic or chaotic, depending on the value of k.

Values of k smaller than 3 give a constant value. Then starts a period doubling with a second bifurcation at 3.5, chaos shortly afterwards, and 3-step period around k=3.8 .

To know more about the logistic equation:

In hebrew, snunit virtual chaos museum has a very nice introduction to this and other chaos themes.

A popular-level explanation of the logistic equation, with an applet, at chaos in the pond.

References to May's original articles can be found in the bibliography of J. Gleick's book "Chaos" (not in the hebrew edition).

Check also the chaos references in my running links.

the logistic equation generator

The logistic equation generator is a simple program for a PC computer, that demonstrates the behavior of the logistic equation for different values of k. The values of x generated by the formula are represented on the screen as colored bars, with sound from the PC speaker. By changing the value of the parameter k, the user can explore the different domains of period doubling, chaos and intermittent order.

The program is written in Qbasic, which comes with newer versions of MS-Dos and can be operated from Windows 95. It is a text file which must be saved on your computer before you can run it.

To get the program, press at the place marked below. You will then see the program text on your browser screen. Use the menu commands "File: Save" to save it as "logistic.bas" in some directory on your computer.

To run the program, find the file you saved in the Windows directory (from "My computer"), and double-click on it. This should launch Qbasic with the logistic generator program. If it doesn't, you can try to launch Qbasic from the Dos directory, and open "logistic.bas" from it.

Once inside Qbasic with "logistic.bas", type "Shift-F5" to start and follow the instructions on the screen.

To get the program, press here.

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