Initially a function f(x) must be created. This entails defining the
function f(x), giving the initial value of x to start the iteration
process, giving the maximum and minimum range you want to look at, and, if
you included parameters a or b in your definition of f(x), you will be
expected to supply initial values for these. For example if you defined
the function ax(1-x) you might choose the initial value of x to be .9,
the minimum y to be 0, the maximum y to be 1, and the initial a to be 2.2.
There are eight major iteration routines available, called Numerical
Iterates, Plot Iterates, Successive Iterates, Graphical Analysis,
Attractors, Density Distribution Analysis, Initial Data Dependence, and
Orbit Diagrams . In the following we use the notation f2(x) = f(f(x)),
f3(x) = f(f(f(x))), etc.
Numerical Iterates. This generates the actual numerical values of f(x),
f2(x), f3(x), ..., starting from your initial x. In the above sample
function the sequence is
.9, .198, .349..., .500..., .549..., .544..., .545...,
which indicates convergence to a number near .545. Changing a to 3.2 gives
the sequence
.9, .288, .656..., ..721..., .642..., .735..., .623..., .751..., .597...,
.769..., .567..., .785, .539..., .795... .
Plot Iterates. This is a plot of fn(x), the nth iterate of f(x), against
n, using your initial value for x. In the above example, the x-axis would
be labelled from 0 to 100 (representing the iteration number), and the y-
axis from 0 to 1 (the minimum and maximum y you selected). The crosses
depict the coordinates (n, fn(x)). For the above sample function it is
easily seen that within a few iterations the successive iterates have
converged to a value between .5 and .6. Successive groups of 100 iterates
can also be displayed. The initial values of x, a, and b, can be changed
and new plots shown. Changing a to 3.2 and repeating this process, gives
an example of period doubling.
Successive Iterates. This is a plot of the initial f(x), where the range
has automatically been selected to be the same as the domain. The initial
value of x is not used. Now, superimposed on this will be f2(x), then
f3(x), and so on. For the above sample function, it is easily seen that
the initial parabola is being flattened more and more, about a point
between .5 and .6. In the Plot Iterates case the initial value of .9 was
used. Here we are looking at the result for "all" initial values between 0
and 1. Changing a to 3.2 and repeating this process, gives another view of
the example of period doubling.
Graphical Analysis. This is a plot of the initial f(x), where the range
has automatically been selected to be the same as the domain, and
superimposed on this is the line y = x. Your initial value is now used to
generate the sequence f2(x), f3(x), etc. using a cobweb-like diagram. For
the above sample function, the cobweb rapidly converges to a number between
.5 and .6. Changing a to 3.2 and repeating this process, gives another
view of the example of period doubling.
Attractors. This plots the "attractors" of the function, i.e. it is a plot
of x (horizontally) vs f(x) (vertically). It starts with your initial a
and x values, iterates 200 times and then plots the next 50 iterated values
on the screen.
Density Distribution Analysis. This plots the "density distribution" using
your initial value. You are asked how many bins you want to use (from 2 to
250). The domain is then split into that number of bins. As each iterate
is computed, it is put into its bin, and the histogram is drawn. This
continues until one of the bins contains 200 entries. After the histogram
is drawn, you can iterate a set of points picked from this distribution.
The computer counts how many points there are in each bin, and totals them.
It then does a single iteration on this many points, where each point is
picked from the original bins in proportion to the height of each bin. The
top half of the screen shows the old histogram, and the bottom half the
iterated histogram. This iterative process can be repeated on the new
histogram. This is used to illustrate the concept of invariance of
measure.
Initial Data Dependence. This is designed to demonstrate "sensitivity to
initial conditions". You will be expected to supply two initial values (x1
and x2) and then these and their iterates are plotted against each other.
Orbit Diagrams. Here, if the function defined contains the parameter a,
you will be asked to specify minimum and maximum for a. Your initial value
of a is ignored. Now you will see a plot of y against a (along the x-
axis). Starting with your initial value for x, the first 200 iterates are
computed but not plotted. Then the next 50 are computed and plotted for
each a. If for the sample function, you choose a to go from 2 to 4, you
can see that at 2.2 the 50 iterates are all at the same point, between 5.
and .6, whereas at 3.2 they go to two different values.