Overview

The basic scheme here is: Select an iterative function from the Scenarios menu. The function is drawn in the graphical iteration, or master, plot to the left of the main applet screen. This plot is labeled with the function's name and defining parameters. Other plots are different views of what is on this master plot.The histogram (right) plot is most closely slaved to the master plot - if you zoom in on the master plot, the histogram zooms to match, etc. Buttons to the left of the master plot open other plots in external windows. Each shows the same function you manipulate on the master plot. Buttons along the top of a plot, manipulate that plot.

The Master Plot

That's a vertical, then horizonal, segment for each L, going from the diagonal, to the green function line, back to the diagonal. The last L is highlighted in magenta. The animation button controls the animation of the first few steps of the iteration. You can turn that off.

Clicking on the main plot, sets a new x0 for all the plots which obey that. (The bifurcation diagram uses its own x0.) On the time series, return map, and bifurcation diagrams, clicking does a fit operation, restoring the window to its original all-data limits.

Dragging a rubberbanded box, zooms in on these plots. Right-clicking or control-clicking zooms out a bit, but never beyond the fit limits.

The zoom buttons above a plot are similar, except that the zoom-in is by a fixed amount, on the center of the plot.

The mousable plots have a mouse tracking window at lower right, showing the coordinate position of the mouse. For example, on the time series, where the horizontal axis is i and the vertical x(i), the mouse tracker looks like Unlike the others, the mouse trackers on the histogram and time series plots read the data they're displaying, instead of the arbitrary coordinate location of the cursor.

The histogram plot zoom always matches the master plot. To make its bars bigger, use the bin button above the histogram. There are no mouse operations on the Kelly plot.

The plot windows are resizeable, but the plots insist on their original aspect ratio.

The external plots tend to slow down operations, so close them when not in use, using the usual window close icon at the upper right of the little window. You can reopen them when you wish.

If an iteration settles to a single value or cycle, the histogram has a smattering of points plus one or more long bars. If the iteration is chaotic, a range of the histogram looks randomly furry.

Kelly Plot

Tent map, s=1.3, equal weight bins.

In a Kelly plot, each square represents a single iterate xi of the series from the master plot. The values are mapped from upper left, left to right across each line, until run out of values, usually before the end of the last line. Unused squares are grey. Colors on the Kelly plot are assigned by "bins". For N bins, there are N-1 numbers defining the bins, because bin 1 means "all values less than or equal to this one get this color, else check bin 2", and so on, until reach bin N-1. Any value greater than that, gets the highest bin color. In general, Kelly plot colors are arbitrary, but for this applet, there's a system. Colors get progressively deeper from the middle bin, using a range of purples for values below the middle, and blues for values above the middle. If the number of bins is odd, there exists a middle bin, and it's white. The list bins button shows the current bin values and their colors. If you change a value here, the binning scheme changes to "Bins: my own" (labeled at bottom of Kelly plot). If you have trouble seeing the values on the dark color backgrounds, highlight the text using the mouse. There is no value for the highest bin - you can't set that value, since it means "anything else".

• Set N Bins, N Drop - Set how many bins/colors you want. Also set the number of values to drop from the beginning of the list, to get past transient behavior. For instance, in the picture above, I could drop 6 to get rid of the pattern-breaking 3-blue, 3-white at the beginning of the plot. If the number of bins is odd, the middle bin is white.
• Equal size - Take the highest and lowest values from the iterate list after N drop, and cut that into N bins equal pieces.
• Equal size, Data-Independent - Split the range 0..1 into N bins equal pieces.
• Equal weight - Attempts to set the bins so that each bin/color has an equal (ish) number of values. This may not be possible, as for example if you have 7 bins and a 2-cyclic iterate list, there will be two bins with almost all the values. But it tries. First N drop are dropped before calculating.
• Mean Centered - Take the iterate values after the dropped, and find the mean and standard deviation. Then working out from the mean, set each bin to the width of a user-supplied factor times the standard deviation. For example, if the mean were 0.5, with standard deviation 0.2, and you entered 0.5 in the menu for "scale stddev *", the bins would be 0.1 wide. For an odd number of bins, the middle bin would then get values 0.45 to 0.55.
• My Bins - Set your own. This menu looks like list bins, except that since you can't set the highest number bin value, it doesn't show it.

Bifurcation Diagram

Logistic map, default bifurcation parameters.

The bifurcation diagram is very different from the others in this collection, and much less driven by changes on the main plot window. It is not drawing the same data in a different format, but rather generating its own, far more data. It's also big and slow, so close it when not in use. The horizontal axis of the bifurcation diagram is s, running the whole valid s-range shown on the main plot. The vertical axis is iterate x-values. This plot is ~500 pixels wide, so for each of 500 s values, it plots N iterates (500 x N). This plot uses its own values for N iterates and starting x0, both set on its own parameters button menu. Changing x0, s, or N on the master plot, has no effect here. Changing the function or composition (comp) does change the bifurcation diagram.

The colors are in rainbow order, red, orange, ..., purple. The same 6 colors are used for any number of highlighted iterates up to 6. For more, it generates a custom rainbow for however many are needed.

Using the default bifurcation parameters (drop first 50), any part of the bifurcation diagram that is vertically thick, should generate some interesting behavior on the other plots. For instance, although most of the action is above s=3.0 on the logistic function, you can see from this picture that there's more than initial transients around s=1.0.

In the really thick areas to the right of s=3.0, you can find some interesting areas by zooming way in and reading the mouse tracker for s values. For example in this zoomed-in closeup, I placed my cursor at the red + mark inside an empty bubble on the diagram, and read s=3.5677 on the mouse tracker. This suggests an interesting value to type in on the parameters menu on the master plot. (The s-slider is no good for precision values.)