Orbits: Bifurcation Diagram Screen

The colors are based on the slope at x where the graph of f_C crosses the line y=x. If that slope is greater than one, the fixed point is repelling, and shown in red. If that slope is between 0 and 1, the fixed point is attracting, and shown in yellow. If that slope is between -1 and 0, the fixed point is again attracting, and shown in pink. If that slope is less than -1, the fixed point is repelling, and shown in purple.

To see points of, say, period 4, the value of "iterate" needs to be changed. To increase the value to 4, press "N" three times; to decrease iterate, type "n". The window ranges can be changed too: Set up the main window to represent [-2.5, 0.25] as follows: press "x" and then "-2.5" (Enter), then "X" followed by "0.25" (Enter). Also change "y" and "Y" to -2.25 and 2.25 respectively. Note that no new picture is drawn; you have to initiate it by pressing "w" (for "write" the new picture). Superimpose a grid on the picture by pressing Alt-G. (Hold the alt key down and press "G"). Do all this and the following picture should appear: (without the labels)

The above picture shows locations of points of period 4. Note that period 2 and fixed points show up, because they do indeed also have period 4. Also note that the tangent bifurcation in the 4th iterate of X*X+C yields a pair of period 4 cycles for X*X+C for C<-1.93... A zoom (discussed below) would show that one of these cycles is attracting for a very small range of C-values; both sets just appear to be repelling (red & purple) for all C-values.

The program offers four ways of displaying periodic points: Let N be the current value of "iterate"; write f ^k to mean the kth iterate of the Current Function.

1. pDIV Show points of period k, where k divides N. (The color [red,yellow, pink, purple] used in the bifurcation diagram is based on the slope where f ^k crosses y=x, where k is the smallest (ie, prime) period of the point)
2. pALL Show points of period k, where k is any number from 1 to N. (color chosen as above)
3. pPRM Show points of prime period N (ie, points with period N and no smaller period) (color chosen as above but now the only iterate being considered here is N)
4. pORG [from an earlier version of the program] like pALL, except the slope of f ^N is used to determine the color. (Note that if k|N, then the absolute value of the slope of f ^k is greater than 1 <=> the absolute value of slope of f ^N is greater than 1, at a point of period k).

Orbits: Drawing the Orbit Diagram in the Bifurcation Diagram Screen

"orbits" has three controls here: "h" determines how many members of the orbit of 0 are calculated BEFORE any of them are plotted (h=1000 by default) and "d" determines how many subsequent members after this are then plotted. (d=120 by default). Finally, "z" is used for the initial point (the critical number 0 in this case). Press "O" to superimpose the orbit diagram on the above picture. You should see the following:

The pink and yellow (attracting) periodic points from the bifurcation diagram have been covered (appropriately) by white in the orbit diagram; regions of chaotic behavior appear, and the locations of repelling cycles of periods 1,2 and 4 are still visible.

Orbits: Zooming Into the Bifurcation and Orbit Diagrams

What follows results from two "stretches" near the so-called period-3 window in the usual orbit diagram; the bifurcation diagram for points of periods less than or equal to 6 are superimposed. You can clearly see the period-doubling bifurcation from 3 to 6, and the attracting cycle in the period five-window to the right. Note the repelling curves passing through "Misiurewicz" points: At such a C-value, the orbit of 0 is eventually periodic, to a repelling cycle; we can see which repelling cycle that is.

 
 

Orbits: Miscellaneous and Details for the Bifurcation Diagram Screen

To draw the orbit of the x (vertical) coordinate of the point at the cursor, press the space bar. Press "g" to make it go continuously. Press the space-bar to pause it. Press "q" to stop it altogether.

"c" clears the screen.

alt-g draws the grid; "(" and ")" set the horizontal and vertical spacing of gridlines.

F5 can be used to change the Current Function from within this part of the program (you don't need to go back to the Main Screen) (remember to use capital X for the variable, and capital C for the parameter).

The Zoom-Factor is set by "Z". By default, it's 2 (Zoom in by a factor of 2 in each direction).

Normally, "orbits" must compute the value of the Nth iterate of the Current Function at each pixel on the vertical strip above each given C-value; that's 480 calculations per C value. To divide the strip into 4 times as many subintervals, set "F" to 0.25. This may catch more crossings of f ^N and the line y=x, at the expense of waiting 4 times longer for the picture to appear.

Normally, "orbits" checks 480 C-values from left to right across the screen. To divide this horizontal interval into 4 times as many subintervals, set "D" to 0.25. This plots more points near period doubling and tangent node bifurcations, again, at the expense of waiting 4 times longer for the picture to appear.

For a fast left to right sketch, use "W" instead of "w". Points are plotted every "G" pixels.

For different coloring schemes, set "K" to different values, from 0 to 8. Enter a value of "K" directly; it also just increments each time you press "K". We'll devote a separate page to describing the schemes; click here for a more complete description.

alt-F8 will cycle through four different built-in palettes, affecting ALL colors in use by the program. These palettes are labeled A, B, C, and D. The palette currently in use is shown at the very bottom left corner of the screen. [the images above were created before this palette listing was added to the program].

• The standard one is (A) and is the one used to produce all the figures in these HTML files. DON'T use it for printing, because you'll waste a lot of black ink and the picture will be awful. Instead, use palette (B).
• (B) is the same as (A) except that black and white are reversed; in particular the background becomes white. Use this palette for printing, in the following way: Once a picture is to your liking, press the "PrintScreen" key on the top row of your keyboard. This "dumps" the graphics image into the Windows clipboard.  Then using a Windows program like "Paint", make a new file from the clipboard's contents and send it off to your printer. Note that yellow may be hard to see; I hope to make the color choices more flexible in a future version.
• (C) and (D) also use a white background but different color schemes for the rest of what's drawn,
• Press alt-F8 to cycle through the list: A-B-C-D-A-B...etc.

Known BUG! If your function f has a vertical asymptote at some value V, the bifurcation diagram algorithm may detect a "crossing" of f with the line y=x when it samples two x values, one on either side of the asymptote V, if f(one x value) is a big positive number and f(the other x value) is a big negative one. What's plotted therefore is the location of the asymptote, not a genuine periodic point. The newer version of "orbits" has a switch which when turned on checks for such a crossing (at the expense of taking a little more time). The switch is "asmpt" and is toggled by pressing Alt-A. Try f_C(X)=1/(X*X+C) for an interesting example; see if you can spot the curve indicating an asymptote instead of a periodic point for f_C.

"=clr" is a switch that toggles when you press "=". just make sure it's off (dark blue) "$zGr" is a switch that toggles when you press "$"; leave it off too(dark blue) . It's for drawing the very hi-resolution pictures found in our CMJ article, meant to be printed on a Canon BJC-4000 series printer. "&Fn" cycles between the user's Current Function (&Fn=0), or two other functions built-in to the program to increase speed. The built-in functions are X*X+C (&Fn=1) and C*sin(X) (&Fn=2). Press "&" to cycle through these three values, but leave it on &Fn=0 to avoid confusion. It's buggy: The little animation window always uses the Current Function, for example.