Parameter Plane and Julia Set Applet

This applet requires the use of a digital signature/certificate in order to save screenshots of graphs generated by the applet. When asked for your approval, you should run/trust the certificate.

Download a copy of Parameter Plane and Julia Set for your computer here and see version history here.

This applet allows the user to see the parameter plane and dynamic plane pictures for several families of functions. The functions one can investigate (via the drop down menu at the top center of the applet) are z^2+c, z^d+c, c*e^z, c*sin(z), c*cos(z), and z^d +c/z^m. It is very similar in use to the Global Complex Iteration Applet for Polynomials and so we explain only the new features here.

Basic Operation

We explain how to use this applet when the family of maps z^2+c is selected, noting that the other families will behave similarly (except for the obvious changes, such as escape criterion). By selecting a value for c, either by typing in the value and clicking the Update button or clicking on the (left) Parameter Plane of c values window, the basin of infinity is then colored in the (right) Dynamic Plane of z values window, with the filled in Julia set colored black. The c value can also be moved around the parameter plane by using the arrow keys on your keyboard.

In the center of the applet we have the following:

• Plot fixed points checkbox which when checked will plot in purple the fixed points of the given map z^2+c (only for quadratic maps).
• Show critical orbit checkbox which when checked will plot the critical orbit. Below the c value is the Period of the cycle (if any) detected by the critical orbit.
• Show orbit 1 and Show orbit 2 checkboxes behave as they do in the Global Complex Iteration Applet for Polynomials.
• Update button will update all pictures and computations based on parameter settings.
• Iterate orbits checkbox allows the user to generate any/all of the Critical orbit, Orbit 1, or Orbit 2 values. This can be done one step at a time or, by typing in a value for n and hitting the + button to generate the orbit values z_1,..., z_n. The points are displayed in the Dynamic Plane of z values, and the values are listed in the tabs at the bottom. (These values can be copied and pasted into another file in the usual way, if one wishes to.)
• Show parameter escape radius and Show dynamic escape radius checkboxes will show the circles (in the parameter plane and dynamic plane, respectively) centered at (0, 0) with the radius prescribed in the Settings tab below.

Under the Settings tab, the user can adjust the following by entering in a new value and then clicking the Update button:

• Parameter plane max iterations value is the number points in the critical orbit that must be checked for escape before a c value in the parameter plane is colored black.
• Parameter plane escape radius determines the escape criterion for the critical orbit when coloring the parameter plane.
• When a new c value is selected by clicking in the Parameter Plane of c values a straight line path of k steps is taken between the old c value and the new c value. At each step, the applet plots the corresponding picture in the dynamic plane. Here k takes on the value listed as the Number of path steps. Also, Delay between path steps is the number of milliseconds the applet pauses before computing the next Julia set.
• Color sample rate affects how drastically or not the shading colors used in the graphs will vary.
• Dynamic plane max iterations value is the number points in the orbit of a z value that must be checked for escape before that z value in the dynamic plane is colored black.
• Dynamic plane escape radius determines the escape criterion when coloring the dynamic plane.
• Dynamic plane min iterations value is the number of iterations made before the escape condition is checked (default is 0).
• Period detection max iterations is the largest number of points in the critical orbit which will be computed when determning the period of a cycle (if any) detected by the critical orbit.
• Period detection tolerance is the value used to determine if a periodic point has been found by the critical orbit. If two points on the critical orbit are within this much of each other, then it is deemed that the critical orbit is converging to a cycle.
• Parameter plane black/white plot and Dynamic plane black/white plot checkboxes will toggle between a color graph and a black/white graph (without needing to click the Update button).

Thumbnail pictures

Thumbnail pictures at the bottom of the applet record previous pictures which can be restored (along with the corresponding applet parameters) by clicking on them. After creating 30 such thumbnails, the first created thumbnail will be overwritten (thus losing the picture that was previously there). The most recently created thumbnail has a white border (red border if picture is in black/white). The last enlarged thumbnail is surrounded by a yellow border. All thumbnails will be deleted when the Delete thumbnails button is pressed or a new family of maps is selected.

Exporting Pictures and Applet Settings

At the top left of the applet are the following drop down menus:

• Export will allow the user to save a png, gif, or jpg picture of the either the whole applet or of the individual viewing windows (Parameter plane or Dynamic plane).
• Colors will allow the user to adjust the color of the text used in the headings of the various sections of the applet.

Note: The family of functions z^w+c = exp(w Log z) + c, where Log z is the principle logarithm for complex constants w and c has been added. The "Julia" set pictures in the dynamic plane represent the split between points with bounded orbit and points with unbounded orbit. The parameter plane represents the split between those c values for which the origin escapes and those for which it does not. However, these maps are not in general analytic and so one must apply the general theory with great caution.

This material is based upon work supported by the National Science Foundation under Grant No. 0632976.