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.

#### Instructions for the Macintosh: Orbit Diagram

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 x0 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