Mathematical Modeling of Social Phenomena

Dr. Courtney Brown

Assignment #9

With this assignment you will be working with the logistic map. This is the model that started all of the fuss about chaos theory (see May 1976 in the bibliography). You might look at this url to see why this model is of interest to scientists studying certain types of populations. The model is X(t+1) = aX(t)[1 - X(t)]. A derivation of this model using population terms can be found at this url. A good url for common definitions useful for the study of chaos can be found here.

Use Phaser to construct the following plots of this model: (1) a plot of X(t) over time, (2) a stair-step diagram, and (3) a bifurcation diagram. Begin by trying a value for parameter "a" of 3.8, with an initial condition of 0.02, and plot the iterations from the first to iteration 100. Explain your results. What do all of those plots represent?

Now try to construct all of the above plots using a value for the parameter "a" that does not produce chaos. Explain your results.

Hand all this in.