- Iterates versus time

- Graphical iteration

We can visualize this orbit in the (*x, y*)-plane by drawing
lines between points on the graph of *f* and points on the line
*y = x*. The following sequence represents the same orbit as
above:

Points of the form (*x*_{n},
*x*_{n}) are on the graph of the identity
function, that is, on the line *y = x*. The points
(*x*_{n}, *x*_{n+1}) =
(*x*_{n},
*f*(*x*_{n})) are on the graph of
*f*.

*Visualizing an orbit with graphical iteration for the Logistic
map*.

Starting with (*x*_{0}, *x*_{0})
on the diagonal, draw a vertical line to the graph of
*f*. Recall that *f*(*x*_{0})
is the next point in the orbit of *x*_{0}. Next, draw a
horizontal line from (*x*_{0},
*f*(*x*_{0})) to the diagonal. This way, we have
repositioned our point so that a vertical line from this new position
will give us *f*(*f*(*x*_{0})). This
procedure, called *graphical iteration*, makes visualization
easier and provides a geometric shortcut; you can even *sketch*
an orbit instead of computing it.

Graphical iteration on the computer can be done with the following software. If you are using a Macintosh computer, follow the instructions for the software Chaos and Dynamics. On the other machines, you should follow the instructions for using Mathematica.

- Use the Java Applet
Iteration of the Logistic Map to see how the dynamics close to the
(positive) fixed points depends on the parameter
. For each
, which point
is attracting? Are there the same number of attracting points for
every ?

- For this question you can either use
Devaney's Chaos Lab #2,
Mathematica or the Java Applet
The Period Doubling Route to Chaos. In the latter case, follow the
abstracted
instructions.

Fix the starting point*x*_{0}= 0.1. For several choices of between 0 and 3.5, investigate the eventual behavior of*x*_{0}= 0.1. In Devaney's Chaos Lab #2 you can push the*Transient*button to see what the eventual behavior is; in the other cases, the eventual behavior is shown in red.

Draw a diagram with on the horizontal axis and*x*on the vertical axis. For each value that you checked, place a dot at the*x*values that are part of the eventual behavior of*x*_{0}= 0.1. If you checked the eventual behavior for enough values, you should be able to sketch the curves connecting the dots, showing the*continuous*dependence on . Do not go beyond = 3.5.

- Iterates versus time

- Graphical iteration

Written by Hinke Osinga

Comments to:
webmaster@geom.umn.edu

Created: Apr 1 1998 ---
Last modified: Thu Apr 9 13:45:03 1998