- 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
*f*(*x*) =*x*(1 -*x*) appears in the frame on the left. The frame gets the title "Quadratic Map", but it means the same.

- The parameter
is called
*a*here. Its value is shown on the right below the first column of options. You can change the value for*a*= by typing in this little window. Make sure that*a*= is between 0 and 4. Press "Return" to see the graph of*f*change accordingly.

- The initial value
*x*_{0}for*x*is shown as the third item in the column on the right. Again, you can change the value for*x*_{0}by typing in this window (*x*_{0}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*x*_{0}, where*k*is the number you typed for*trans*.

- In order to see the graph of
*f*^{2}, the second iterate of*f*, change the number for*nf*in the right column. The Java applet shows the graph of the 2^{nf}-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.

Written by Hinke Osinga

Comments to:
webmaster@geom.umn.edu

Created: Apr 20 1997 ---
Last modified: Wed Apr 15 20:11:33 1998