Bifurcations of the cubic map.
This applet was created by Yakov Shapiro for Professor Robert Devaney.
How to use this applet.
The applet draws a bifurcation diagram for the cubic map f(x)=Cx(1 - x^2/3) . The C-axis is horizontal, and the x-axis is vertical. For each value of C it performs a certain number of iterations (100 by default) of the critical point x=1, without displaying them on the screen. It then performs an additional number of iterations (50 by default) and plots the results against the corresponding value of C. Thus if there is an attracting periodic orbit for some value of C, that orbit will be plotted on the screen.
Changing the number of iterations.
To change the initial and/or the additional number of iterations, enter the new numbers in the text fields in the line "I'm hiding first # iterations of critical point and displaying next ## iterations." Then click "Change". Note that increasing the total number of iterations may cause the applet to run slower.
There are two ways to zoom in. The first is to enter the new numbers in the text fields displaying the range of x and C, and then click "Change". The other way is to press down the left mouse button at one corner of the region you want to zoom to, and then drag the mouse to select the region and release it at the opposite corner. You still have to click "Change" to see the new diagram after you select the region for zooming in.
To zoom out, enter the desired boundary of the region in the text fields for the range of x and C, then click "Change". Zooming out by mouse is currently not supported. However, you can click the "Original diagram" button to view the entire bifurcation diagram. (Another way to do it is to reload the web page containing the applet.)
Displaying the orbit of the critical point.
To see the orbit of the critical point x=1 on the screen for a given C, enter a value of C in the text field marked "Orbit of critical point for C= #", and then click "Display orbit". This will also display the orbit of the critical point in the list on the right. Another way to do this is to double-click a point with the desired value of C in the main screen. The orbit plotted will always be the orbit of the critical point, no matter what value of x is double-clicked.
When the orbit of the critical point is displayed on the screen, the applet will again hide some number of iterations (the same as for plotting the diagram) and then display some more (again, the same as for the diagram). However, the entire orbit will appear in the list on the right.
To remove the orbit from the screen, click the "Clear orbit" button.