To start up Mathematica in a Unix window, type 'math', or 'mathematica' if your prefer the notebook version. On a PC, use the start button and find Mathematica in the Program directory. The Package written for this course is called 'Chaos.m'. Inside Mathematica, load this Package by typing

When you load the Package, the file 'Logistic.m' is loaded automatically, so that the current function is set to the Logistic map

*Function Iteration with
= 2.*

- Upon loading the Package, there are some messages printed. The
last line says

*lambda = 2, x0 = 0.94, its = 100*

These are the variables and parameters that you can change. If you type

*lambda = 3.1;*

the parameter is set to 3.1. Similarly*x0*is the initial value for which the orbit is computed, and*its*is the length of the computed orbit, that is, the number of iterates. (Note that*x0*should be chosen in the closed interval [0,1].)

- Next, type

*PlotIterates*

A figure will appear with the plot of the iterates versus time.

- You may wish to Look at only part of this picture, especially when
the number of iterates is large. Notice that the figure has a number,
for example

*Out[5]*

If you type

*Show[Out[5], PlotRange -> {{50, 100}, {0, 1}}]*

A new figure appears with only the 50-th to the 100-th iterates. You can use the shortcuts*%*and*%%*for the numbers of the last and next to last output lines.

- Experiment with different values for
and
*x*._{0}

To start up Mathematica in a Unix window, type 'math', or 'mathematica' if your prefer the notebook version. On a PC, use the start button and find Mathematica in the Program directory. The Package written for this course is called 'Chaos.m'. Inside Mathematica, load this Package by typing

For the Logistic map, pick a value between 0 and 4, and iterate as follows:

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

- You can change the variable
*x0*and parameters*lambda*and*its*by typing something like

*x0 = 0.12345;*

- Next, type

*PlotGraphical*

A figure will appear with the graph of the Logistic map for this specific value. The line*y = x*is drawn as well, and you see the graphical iteration of*x0*with the last iterates drawn in red. The actual points of the eventual behavior are drawn as thick red dots on the diagonal.

- You may wish to zoom in to this picture. For this, type

*Show[%, PlotRange -> {{0.6, 0.8}, {0.6, 0.8}}]*

A new figure appears of only the box [0.6, 0.8] X [0.6, 0.8]. You can type something similar to zoom in more, but also to zoom out, even though the last picture does not show all points.

Written by Hinke Osinga

Comments to:
webmaster@geom.umn.edu

Created: Apr 3 1998 ---
Last modified: Mon May 11 16:35:12 1998