# 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 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 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.