A great deal of chaotic behavior can be described by one, simple, fairly innocuous looking equation, the logistic map. The logistic map is recursive, meaning that the third term is a function of the second, the fourth a function of the third and so on. This is written as;
xn+1 = m xn(1 - xn)
where m is some constant, a constant which we shall see becomes very, very important. Implementing this function in a programming routine is very simple (source code), the interpretation of the results and using computer programs to further investigate chaotic behavior are not, for most people James Gleick's book Chaos is a great starting point. As for any other physics students out there, you have more than enough resources to get started, find your own way.
In order to explore the behavior of the logistic map, I ran my C++ routine with various values for the constant m, dumped the output into an ascii file, and then graphed the output using gnu plot. Windows users may have an easier time importing the data into Excel and then graphing from there. The first value I attempted was m = 2.8, the output being a very old and recognizable friend to the physics or engineering student, a damped harmonic oscillator;
It is perhaps best to think of the logistic map as nothing more than the equation for an harmonic oscillator. The harmonic oscillator describes a wide array of physical phenomena, from such simple things as a weight on a spring, or a pendulum, to the more exotic, such as an electron bound to an atomic nucleus. The above would be the equivalent of an oscillator whose motion damps out over time, or a pendulum which swings more and more slowly over time finally coming to a complete stop. Further graphs of the logistic map will continue to show this close relationship to the harmonic oscillator, and in fact a driven pendulum, a very common object, is known to show chaotic behavior.
For m = 3.3 the behavior is exactly that as for an oscillator with no damping, this oscillator will continue to run indefinitely;
Now for values of m greater than 3.3 the logistic map starts to show chaotic behavior. By chaotic behavior I mean that even though the equation is straightforward and deterministic, its behavior is not. Not any clearer? Looking at the diagram below you will see that different peaks have different shapes, and in fact this equation will give different shapes for any value of xn, specifically as the equation maps out, each "piece" will look different from those which precede it. This equation has lost its periodic recognizability.
A close-up of the above shows in greater detail how each peak is different from those which precede it;
This is the type of behavior which scientists and mathematicians classify as chaotic. For more normal functions, each part of the curve will have some definite shape, clearly related to functions of the same type, but perhaps steeper or slightly off if one of the parameters is changed in a small way. However in what we have seen above a small change, namely the increase of m from 3.3 to 3.8 has resulted in a drastic, definitely chaotic change in the behavior of the function. For a deeper look into that behavior, continue on to the bifurcation diagram.