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Instructions for the Java Applet
The Period Doubling Route to Chaos
If you click on the link below, you get to the website of Michael
Cross about the Period Doubling Route to Chaos. Between two horizontal
lines, you will find "click here to start application" and next to it
a button with "Start" on it. Click on "Start" to load the Java
applet. The Java applet appears in a separate window. You can click
"Back" to continue reading this page.
- Start the Java applet:
The Period Doubling Route to Chaos. This
Java applet has a lot more possibilities than are discussed in the
One-Dimensional Dynamical Systems Lab, and we only give instructions
relevant for this course.
- A separate window appears with buttons and numbers. Click on the
"Reset" button so that the graph of the Logistic map
(1 - x) appears in the frame on the left. The frame gets
the title "Quadratic Map", but it means the same.
- The parameter
a here. Its value is shown on the right below the first
column of options. You can change the value for a =
in this little window. Make sure that a =
0 and 4. Press "Return" to see the graph of f change
- The initial value x0 for x is shown
as the third item in the column on the right. Again, you can change
the value for x0 by typing in this window
(x0 must be between 0 and 1). Press "Return" to
see the blue vertical line change accordingly.
- There are two ways to iterate. If you press "Start", the fourth
item in the left column, the iteration starts and continues until you
press "Stop". The speed can be controlled with the scroll bar in the
lower left corner. The author warns for bugs when using high speed.
If you press "Step" the iteration will be done stepwise, waiting for
you to press "Step" again when you want to see the next iteration.
- If you press "Clear", the Java applet removes the transients. When
you hit "Start", you can clear without pressing "Stop" first. Press
"Start" to continue the iteration. If you press "Reset", the Java
applet returns to the initial state. The changes you made by typing
will stay the same.
- Another way to remove transients is by changing the number for
trans, the last item in the right column. After pressing
"Return", the vertical line will be placed at the k-th
iterate of x0, where k is the number you
typed for trans.
- In order to see the graph of f2, the second
iterate of f, change the number for nf in the right
column. The Java applet shows the graph of the 2nf-th
iterate. So nf = 0 is the first iterate, nf = 1 the
second, and nf = 2 the fourth already. It is not possible to
see the third iteration. It is also not possible to see two graphs at
the same time.
- To stop the Java applet, kill the applet window.
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The Geometry Center Home Page
Written by Hinke Osinga
Created: Apr 20 1997 ---
Last modified: Wed Apr 15 20:11:33 1998