**Up:** *Linear and Nonlinear Behavior*

# One-Dimensional Dynamical Systems

## Part 4: Linear and Nonlinear Behavior

#### Instructions for the Macintosh: Graphs of higher iterates

The software Chaos and Dynamics can be found in the apple-menu, under
'Math Apps'. Go to 'Devaney's Chaos Lab' and load
'Lab 2-Plot and Analyze'.
#### Instructions to graph higher iterates

- Push the
*Add Graph* button, marked with a '+'. Now pick a
value for .

- Push the
*Add Graph* button again, and change the value in
the *Iterate* column from 1 to 2. (Make sure that both graphs
correspond to the same
.)

- To get a closer look at a region, use the mouse to select a
region, and click on the
*Magnify* button.

The software Chaos and Dynamics can be found in the apple-menu, under
'Math Apps'. Go to 'Devaney's Chaos Lab' and load
'Lab 3-Orbit Diagram'.

*An enlargement of the orbit diagram for the Logistic family.*

#### How to Study the Orbit Diagram

- To start the computation, click on the
*Iterate* button.
The program begins by iterating an arbitrary point
*x*_{0} as many times as specified in the
*Iterations* box. However, iterates are displayed only after the
number of iterates in the *First Shown* box. Therefore, the
*transients* are not shown in the picture.

- To get a picture that reflects the eventual attracting behavior
more accurately, increase the numbers in the
*First Shown* and
*Iterations* boxes.

- To look at a particular area, select the area with your mouse, and
click on the
*Magnify* button. Notice that the numbers in the
*First Shown* and *Iterations* boxes are increased
automatically. You can still increase these numbers even more, if you
like.

**Up:** *Linear and Nonlinear Behavior*

*The Geometry Center Home Page*
Written by Hinke Osinga

Comments to:
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Created: Apr 3 1998 ---
Last modified: Wed Apr 8 19:24:53 1998