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| Henon Attractor |
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Henon Attractor
- This image is a variation of the Henon Attractor, which is a fractal in the division of the chaotic Strange Attractors named after astronomer Michel Henon. The Henon Attractor emerged from Henon's attempt to model the orbits of celestial objects.
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Basic Description
The Henon Attractor is a special kind of fractal that belongs in a group called Strange Attractors, a category of images that is very strange indeed.
A characteristic of this strange function is that it drawn irregularly. If you iterate the functions that describe the Henon Attractor and plot the points of the functions for each iteration in a time sequence, you would observe that the points jump from one random location within the image to another. If you take a look at the animation, you can see the irregularity of the first 120,000 plotted points.
Also, this image is an iterated function, meaning that the equations that describe it can be repeated an infinite amount of times. In fact, if you magnify this image, you would find that the lines that appear to be single lines on the larger image are actually sets or bundles of lines, who, if magnified closer, are bundles of lines and so on. This property is called self-similarity, which means that even as you look closer and closer into the image, it continues to look the same (see the image to the left). In other words, the larger view of the image is similar to a magnified part of the image.
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A More Mathematical Explanation
- Note: understanding of this explanation requires: *Algebra
This image is a modified version of the famous Henon Attractor that will be described below.
Chaotic System
The Henon system can be described as chaotic and random- its function does not plot points that go to infinity not does it plot points in a repetitive pattern. However, the system does have a structure in that its points settle very close to an underlying pattern called a chaotic attractor or basin of attraction. The Henon Attractor can be described by the following equations:
The Henon Attractor uses the values a = 1.4 and b = 0.3, creating the image shown to the left.
The artistic image of the Henon Attractor that is the subject of this page instead uses the values a = 1 and b = 0.542. Henon Attractors usually begin with a starting point (1,1).
Fractal
The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in the another direction. A Cantor Set can be simply described as a iterated function beginning as a line segment that is divided into three segments, whereupon the middle segment is removed and the end two lines become the next line segments for the iterated method.
The Fractal Dimension of the Henon Attractor is not calculable due to its chaotic behavior, but it is estimated to be about 1.261.
Fixed Points
As seen from the previous system of equations, the Henon Attractor uses only two variables (x and y) that are evaluated into themselves, which results in two equilibrium or fixed points for the attractor. This points are very unique, because if the system ever plotted onto the fixed points, the plotted values would no longer change and would remain at the fixed points.
The two fixed points of the Henon Attractor must satisfy 
and 
.
Using the Henon Attractor's system of equations , the fixed points are (0.6314 , 0.1894) and (-1.1314 , -0.3394).
![\frac{1 - b}{2} \pm \left[(\frac{1 - b)^2}{4} + a \right]^{1/2}](/mathimages/imgUpload/math/0/0/6/00676e967a8f0007eb57b0bffb93db0f.png)
About the Creator of this Image
The images created by this author were found on the author's (username SiMet) Picasa Web Album under the category "Computer Art".
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