The next logical step is to extend the concepts of iterated systems to multiple-dimensional mappings. Given the results of the previous explorations one can assume that much of the behavior found in the quadratic map will have its analog in higher dimensions and thus we need not introduce an entirely new vocabulary.
Let
be a mapping of ordered n-tuples of real numbers on to themselves. Take the results and feed them back into the map repeatedly. For convenience sake, let's call the n-tuples points and Rn a space of dimension "n". This procedure generates an orbit of points
in our space from a seed "p". The behavior of orbits in higher dimensions is similar to that in one dimension. Let's look at some of the possibilities in two dimensions.
It's quite easy to devise mappings that illustrate attracting and repelling fixed points. For example,
draws all points asymptotically towards the origin while
drives them away to infinity. The mapping
will set every point on its own four-cycle around the origin. Generating a mapping that produces its own unique n-cycle was a bit more difficult. One candidate
seems to send seeds on to an 11-cycle, but I'm not quite sure. This area of mathematics is highly dependent on computing and I have no computer program for generating arbitrary two-dimensional iterated mappings at my disposal (although I manage it on a programmable calculator).
In higher dimensions, however, attraction and repulsion are not limited to points. An iterative map can collapse on to any structure possible in that dimension. Attractors and repellers can form paths, surfaces, volumes, and their higher dimensional analogs. For example, the two-dimensional map
attracts all points asymptotically to the x-axis. Likewise, a two-dimensional object can act as a repeller. Such is the case for the map
Points inside the unit circle head for the origin while those outside fly off to infinity. Points on the circle remain there and thus (for this map) the unit circle can be considered a fixed repeller.
For comparison, take the set of iterated functions
where "a" and "b" are constants set at 1.4 and 0.3 respectively. Those seed values that do not escape to infinity collapse on to the bizarre creature shown below. This is an example of a strange attractor; the Henon attractor, named after its discoverer, Michel Henon. Although composed of lines, orbits on this beast do not flow continuously, but hop from one location to another. When drawn, the Henon attractor seems to materialize out of nothing. It is also chaotic. All seed values that converge to the attractor do so in a different manner. Distinct points that are initially separated by even the most minuscule gap will eventually diverge and evolve separately. The Henon attractor also shows a great deal of fine structure (an infinite amount to be exact). Successive magnifications show an ever increasing degree of detail. Any cross-section through an arm of the Henon attractor is equivalent to a Cantor middle thirds set.
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x = [-0.4046, +0.4010] y = [-1.3490, +1.3370] |
x = [0.1500, 0.2500] y = [0.2600, 0.3600] |
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x = [0.2070, 0.2200] y = [0.3020, 0.3100] |
x = [0.2103, 0.2110] y = [0.3050, 0.3055] |
There are dozens of well-studied strange attractors. A few examples are shown below. The most famous is without a doubt the Lorenz attractor. This attractor emerged from the study of equations used in the prediction of weather. It was here, sometime in the early '60s, that chaos was first seen.
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Ikeda (2D) | Ushiki (2D) |
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This page is a part of E-World! Maintained by Glenn Elert Last modified Saturday, 21-Feb-1998 15:58:08 EST Go to the nearest index |