## Java 1.1 Applet for Iterating the Logistic Equation

This
applet can be used to experiment with iterations of the logistic equation
*x*_{k+1} = Q(a, x_{k}) = a x_{k}(1 -
x_{k}), k = 0, 1, 2, ...

The initial guess
*x*_{0} and the parameter *a* satisfy the conditions
*0 <= x*_{0} <= 1, and *0 <= a <=
4*

For values in these ranges the iterations remain bounded and
satisfy *0 <= x*_{n} <= 1 for all *n*. We are
interested in the long term behavior of the sequence *x*_{0},
x_{1}, x_{2}, ..., x_{k}, ... for given values of
*a* and *x*_{0}.
The applet also graphically shows the iterations using a staricase and cobweb
diagram. The graph shows the parabola *y = ax(1 - x)* and the line
*y=x*. The sequence of iterations is shown as a sequence of lines
(vertically to the parabola horizontally to the line *y=x*, vertically to
the parabola, horizontally to the line *y=x*, and so on). The *x*
values of the vertical lines correspond to the terms in the sequence. For
monotone convergence to a limit we have a staircase diagram (try
*a=1.9*), for oscillating convergence to a limit we have a cobweb diagram
(try *a=2.9*). For values of *a > 3.0* there is no convergence.
Either a periodic point is obtained or chaos is obtained. Try various values of
*a* and analyze the behavior of the logistic equation. For the long term
behavior it is best to skip a lot of iterations before plotting them. The actual
values of the last few iterations are shown.

### Other Applet Parameters

*order*
- If order is left at its default value of
*1* then the sequence
generated is *x*_{k}, where *k=0,1,2*,.... If order is
set to *2* then the sequence generated is *x*_{k}, where
*k=0,2,4,6*,..., i.e., every other iteration. In general if order is
set to *m* then the sequence generated is *x*_{k}, where
*k=0,m,2m,3m*,...
*iterations to skip*
- if this value is set to
*s* then the iterations
*x*_{k} are calculated but not displayed until *k=s*.
This is useful for analyzing the long term behavior.
*iterations to display*
- this is the number of iterations to display beginning with iteration
*x*_{s}.