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 [5]
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.