INTRODUCTION TO CHAOS:

A SIMPLE CASE OF CHAOTIC DYNAMICS

D. C. Mikulecky

The simplest example of chaotic dynamics arises in the case of the *logistic equation
*which has the following motivation. Imagine a growth process in ecology, economics, or
some other related growth process. The system is monitored in discrete time steps, e. g.
generations, years, etc. Let X_{n} represent the population size in the nth
generation. The most naive growth law would be that the next generation depends on this
one in a proportional way, with the growth factor, r, a being the constant of
proportionality.

X_{n+1} = rX_{n}.

A somewhat more sophisticated model says that, if unchecked, the growth will follow an exponential law,

X_{n+1} = r^{k}X_{n}.

If r > 1 , X tends towards infinity.

A more realistic model recognizes that growth is limited by limited resources,
space,etc and that these impediments also increase with population. As X_{n} tends
to get large, the growth rate tends to slow down. The simplest way to incorporate this is
to let r = A(1-X_{n}). Then the expression is

X_{n+1} = A(1-X_{n})X_{n}.

With this model one can calculate the growth of this population on a hand calculator.
If you try it, you must supply a numerical value of the growth parameter, A, and some
initial population size, X_{n} (0 < X_{n} < 1).

Another way to make the calculation is by a graphical method. The figure below illustrates this method.

Figure 1: The "phase portrait" for the logistic equation in the stationary realm.

The equation, which describes a parabola, is plotted in figure 1 with the
particular value for A being set at 2.5. This determines the exact shape of the parabola,
which is of crucial importance. The line at 45 degrees is the locus of all points for
which X_{n+1} = X_{n}. By starting with the initial value on the
horizontal axis (vertical line with arrow), the new value is found by drawing a verticle
line to the parabola. [On your calculator this is equivalent to multiplying the initial
value of X_{n} by (1 - X_{n} )A]. Project the point of intersection
horizontally to the vertical axis to determine X_{n+1}. Then, to do the next
iteration, extend this horizontal line until it intersects the 45^{o} line. This
converts X_{n+1} into X_{n} for the next iteration. For this value of A
(2.5), the process quickly converges on a particular stationary vale of X as can be seen
on the graph as a set of ever diminishing partial rectangles. This final point represents
the **equilibrium for the system. **[On your calculator, this value will **always **result
in itself being calculated for X_{n+1} when it is fed in as X_{n.}]_{
}

In graphical form to the left we see the successive values of Xi plotted by generation, i. After an initial transient phase, the system settles into a constant value in time.

_{The next example is for a slightly higher value of A and it
results in periodic behavior.}