From Math Images
Basic DescriptionThe Henon Attractor is a special kind of fractal that belongs in a group called Strange Attractors. It is plotted in an irregularly matter, and can be modeled by two general equations. The Henon Attractor is created by applying this system of equations to a starting value over and over again and graphing each result.
Making the Henon Attractor
Say we took a single starting point (x,y) and plotted it on a graph. Then, we applied the two Henon Attractor equations to the initial point and emerged with a new point that we graphed. Next, we took this new point and again applied the two equations to it and graphed the next new point. If we continued to apply the two equations to each new point in a process called iteration and plotted every outcome from this iteration, we would create a Henon Attractor.
Furthermore, if we plotted each outcome 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 the 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, that, 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 Henon Attractor is often described as being similar to the Cantor Set. Let us zoom into the Henon Attractor near the doubled-tip of the fractal (as seen in the animation). We can see that as we continue to magnify the lines that form the structure of the Henon Attractor, these lines become layers of increasingly deteriorating lines that appear to resemble the Canter Set.
The Fractal Dimension of the Henon Attractor is not calculable using a single equation, but it is estimated to be about 1.261.
Changing "a" and "b"
Here are some more examples of Henon Attractors with different a and b values.
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 two Henon Attractor equations are applied to the fixed points, the resulting points would be the same fixed points. In algebraic terms:
- where is the x-value at the nth iteration and is the x-value at the next iteration.
Therefore, if the system ever plotted onto the fixed points, the fractal would become stagnant. By solving the Henon Attractor's system of equations with a = 1.4 and b = 0.3, we can find that the fixed points for the original Henon Attractor 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 stable. This means that if a point is plotted close to the fixed point, the next iterated points will remain close to the fixed point. The second fixed point (-1.1314 , -0.3394), labeled "2", is considered unstable, and it is located outside of the bounds of the attractor. An unstable fixed point is such that if the system gets close to the fixed point, the next iterated points rapidly move away from the fixed point.
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About the Creator of this Image
Piecewise Affine Dynamics is a wiki site that was created by a group of French mathematicians that is dedicated to providing information about "dynamic systems defined by piecewise affine transformations".
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
Future Directions for this Page
A better, less vague description of how sections of the Henon Attractor resembles the Cantor Set
Also, the description of the Henon Attractor can be expanded to include a discussion about the fractal's "basin of attraction". For more information, click here.
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