The Logistic Map:
This is one of the simplist examples of a chaotic system, yet it reveals much about the structure and causes of chaos. The simplist form of the Logistic map is given by the sequence equation:
One can plot the results of this sequence with a "Cobweb" diagram. If we plot the logistic equation above, with the left hand side identified with the y axis, we will have a curve in the x-y plane whose position relative to the x=y line is determined by the control parameter a. A Cobweb diagram traces out the sequence created by the Logistic map as follows: we begin at the initial condition along the x axis. This is used to seed the mapping and produce our next number, which is on the y-axis and is located by the logistic function curve. Thus we draw a verticle line from the x-axis to the logistic curve. In our next iteration, this number acts as our x-axis number so we jump to the x=y line and back up to the logistic curve identifying the y result, repeating the process. The animation below shows such a diagram, with initial x at -1.4, as we progress the control variable a from -0.25 to 2:
As we can see, the Logistic mapping passes from order to chaos, but has interesting windows of order beyond the turn over, a temporary silencing of the chaos.
This is clearly demonstrated by an orbit diagram (famously known as a bifurcation diagram). In this diagram, we itterate the logistic map many times at each point along the control parameter. A period two orbit, for example, would be an oscillation back and forth between two points, and would show up as only the manifestation of two points above the control paramter's position along the x-axis. If it is chaotic, there will be many such points.
An interesting feature about the orbit (bifurcation) diagram is that it is self similar. One can see this very clearly in the following animation which zooms in on three areas of self similarity, though one could go on ad infinitum:
One can find a cool Java applet for investigating the bifurcation diagram here.
Further Investigations on the Logistic Map
The Wolfram site goes into greater mathematical detail.
Wikipedia, an excellent and GNU encyclopedia, has a fully hypertexted entry here.
References
- R. Gilmore and M. Lefranc, The Topology of Chaos, Alice in Stretch and Squeezeland, NY: Wiley, 2002.
- S. Strogatz, Nonlinear Dynamics and Chaos, Westview Press, 1994