10182

Orbits of the Hopalong Map

This Demonstration plots a number of orbits of the two-dimensional iterative map known as the Hopalong map. The Hopalong map can be represented by the recursion equations: , , with and ranging over the reals and equal to either 0 or +1.
Different orbits are generated using different initial points defined by the "" locators. Initially, 25 orbits are plotted. You can drag the initial points or add or delete new ones (Alt+Click on Windows, Command+Click on Macintosh) inside the plot. Alternatively, up to 50 random initial points can be launched.
The map parameters , , and can be varied manually or a random set can be used. Use the "range" slider as a magnifier to see more detail inside some orbits. Check "full range" to see all points from all orbits.
The Hopalong map is effectively a simplified version of Martin's map (see the Demonstration "Orbits of Martin's Map"): {, }. By dividing both sides of the equations by , as suggested by Barry Martin, we obtain: , with either 0 or +1. This way, only two parameters are needed and we can divide all possible orbits into two classes with either 0 or 1.

DETAILS

The Martin map equations were taken from [1] and [2].
The simplified, two-parameter equations for the map were suggested to the author by Barry Martin (see his Torinohop website).
The Martin map is also called the "Hopalong-Attractor".
References
[1] M. Trott, The Mathematica GuideBook for Programming, New York: Springer–Verlag, 2004 pp. 347–349.
[2] B. Martin, "Graphic Potential of Recursive Functions," in Computers in Art, Design and Animation (J. Landsdown and R. A. Earnshaw, eds.), New York: Springer–Verlag, 1989 pp. 109–129.

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