Abstract
The Macintosh application StdMap allows easy exploration of many of the phenomena of area-preserving mappings. This tutorial explains some of these phenomena and presents a number of simple experiments centered on the use of this program.
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For the purposes of this introduction, if you have ever used StdMap before, you should delete the preference file :_/Library/Preferences/edu.colorado.stdmap.plist, so that you can start the application in its default state.
For the significance of this parameter value, see §7.
On my 3 GHz machine, N is 91074 to give an elapsed time for N iterations of about 1/60-second. You can view or change this using the Change menu and selecting Speedometer....
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The second way to slow iteration is to decrease the global variable N that represents the number of iterates done for each function call when you are in continuous iteration mode, ⌘-G. You can do this with Change → Speedometer.
It is a standard Macintosh convention that drawing is constrained if you hold down the shift key. In this case, the line will be constrained to be exactly vertical, horizontal or at forty-five degrees. Constrained drawing is also an easy way to draw squares and circles.
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In StdMap, the Suris map has a second term πb sin(2πx) that destroys integrability when b ≠ 0. Type ⌥-⌘-1 to access it; here ⌥-is the option key — hold this key down and then type ⌘-1. This map was studied in H E Lomelí and J D Meiss, Phys. Lett. A269(5–6), 309 (2000)[20].
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The initial condition dialog has a pop-up menu labeled symmetry. The default setting for this is minimize, and this gives the orbit above. Choosing minimax from this menu will give the elliptic (2,5) orbit. The symmetries of periodic orbits will be discussed in §6.
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There is also a pop-up menu for selecting the symmetry of the orbit. Leave this set at the default minimizing as this corresponds to the hyperbolic saddle.
We allocate an array that is no more than half the physical memory in your computer and no larger than needed to contain up to 10 times the number of pixels in the plot window.
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Do not set a = 1 for the Suris map, as in this case F(x) = 2x and every point is parabolic and has period-4. The Suris map approximates the standard map with a ≈ k/4, so try a = 1/4 to obtain a picture similar to the left pane of figure 7.
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Here we are really considering the lift of the map to the plane, by undoing the mod 1 operation. It is convenient to do this to count the number of rotations of an orbit. Indeed, as was suggested to me by Robert MacKay, StdMap uses a structure, ifnumber, for x containing a long and double for the integer and fractional parts.
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When k < 0 the dominant line for (1) is Fix(S 3). Other reversible maps, like the Hénon map, ⌘-2, also appear to have dominant symmetry lines, though as far as I know this has not been proved.
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Near, e.g., (0.306, 0.612) when k = 1.5.
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It would be best, of course, for me to add a parser to StdMap so that arbitrary maps could be iterated. However, much of the code depends upon the reversibility assumption, so this would not be easy.
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Dedicated to the memory of John Greene. His ideas are as timeless as his generosity is legendary.
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Meiss, J.D. Visual explorations of dynamics: The standard map. Pramana - J Phys 70, 965–988 (2008). https://doi.org/10.1007/s12043-008-0103-3
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DOI: https://doi.org/10.1007/s12043-008-0103-3