From Math Images
The Henon Attractor is a special kind of fractal that belongs in a group called Strange Attractors.
A characteristic of this strange fractal is that it is drawn irregularly. The Henon Attractor is described by two equations. Let us say that we take a starting value (x,y) and apply the equations to the starting values and then the resulting outcome over and over (a process called iteration). If we plot every outcome from this iteration one at a time, we would observe that the points jump from one random location to another within the image. If you take a look at the animation, you can see the irregularity of a number of plotted points. Eventually, the individual points become so numerous that they appear to form lines and an image emerges.
This image results from an iterated function, meaning that the equations that describe it can be applied to itself an infinite amount of times. In fact, if you magnify this image, you would find that the lines (really many, many points) 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. In other words, the larger view of the image is similar to a magnified part of the image.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Algebra
The image featured at the top of this page is a modified version of the Henon Attractor that will be described below.
The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in another direction. If we zoom into the Henon Attractor near the doubled-tip of the fractal (as seen in the animation), we can see that the points form layers of lines that appear to resemble the Canter Set. If we follow the attractor backwards from the doubled-tip, we can see that the fractal is more smooth and contains less bundles of lines.The Fractal Dimension of the Henon Attractor is not calculable using a single equation, but it is estimated to be about 1.261.
The Henon system can be described as chaotic and random. However, the system does have 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 and begin with a starting point (1,1).
Looking at the system of equations that describe the fractal, the Henon Attractor uses only two variables (x and y) that are evaluated into themselves. This results in two equilibrium or fixed points for the attractor. Fixed points are such that if the system of equations are applied to the fixed points, the resulting output would be the same fixed points. Therefore, if the system ever plotted onto the fixed points, the fractal would become stagnant.
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).
To solve the system of equations:
If and then
Using the quadratic equation
Using a = 1.4, b = 0.3:
Using y = bx:
There are two types of fixed points, stable and unstable. The first fixed point (0.6314, 0.1894), labeled "1" on the image, is located within the bounds of the attractor and is unstable. This means that if the system gets close to the point, it will exponentially move away from the fixed point to continue chaotically. The second fixed point, labeled "2", is considered stable, and it is located outside of the bounds of the attractor.
How the Main Image RelatesThe artistic image of the Henon Attractor that is the subject of this page instead uses the values a = 1 and b = 0.542.
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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".
Glenn Elert, The Chaos Hypertextbook Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and fractals
Bill Casselman, Simple Chaos-The Hénon Map
www.ibiblio.org Henon Strange Attractors
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