One of the truly amazing aspects about chaotic behavior is that it can arise from very simple systems. We have already mentioned the strange attractor generated from the three-dimensional Lorenz equations (5). In this section we will study a variety of one- and two-dimensional maps from which one can obtain chaos.
This famous mapping from biology was first studied by R. May and describes the change in an animal population from one year to the next. The basic equation is the following simple one-dimensional map 
where is the population for the year and a is a constant. The first term in the above equation gives unrestricted Malthusian growth, while the second term, from Verhulst, models factors which limit the population (competition for food, predators, etc.). Note that has been suitably scaled so that it always lies between 0 and 1. Write a script for following the iterations of the map for each value of a. You will also need scripts for displaying a table of values and for plotting as a function of n. We want to study the ``steady-state'' behavior of this system for moderately large n values. That is, for each a value, examine what happens to the value as n is increased. As n gets large, does approach a single limit, or are there two (or more!) limiting values?
First, study the behavior for the following values of a:
You see, for the first case, that the population cannot sustain itself. Are the results sensitive to the choice of the initial x value? How many iterations are required to give reasonable convergence? What is the (subtle) difference between the second and third cases? The behavior so far is for a single limiting value as n increases indefinitely.