Henon Attractor

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|Image=HenonMain.jpg
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|ImageIntro=This image is a Henon Attractor (named after astronomer and mathematician Michel Henon), which is a fractal in the division of the chaotic strange attractor.
|ImageIntro=This image is a Henon Attractor (named after astronomer and mathematician Michel Henon), which is a fractal in the division of the chaotic strange attractor.
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|ImageDescElem=The Henon Attractor is a special kind of fractal that belongs in a group called [[Strange Attractors]], 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.
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The Henon Attractor is a special kind of fractal that belongs in a group called [[Strange Attractors]], 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.
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===Magnification of the Henon Attractor===
===Magnification of the Henon Attractor===
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|ImageDesc===Fractal Properties==
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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.
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===History of the Henon Attractor===
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Michel Henon was a French mathematician and astronomer who developed the Henon Attractor in the 1970s. At that time, Henon was interested in dynamical systems and especially the complicated orbits of celestial objects. The Henon Attractor emerged from Henon's attempt to model the chaotic orbits of celestial objects (like stars) in the mist of a gravitational force.
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==Fractal Properties==
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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 attractor itself 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.
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 attractor itself 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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|other=Algebra
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|Field=Dynamic Systems
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|References=:*Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook]
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Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook]
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Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals'''
Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals'''
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Bill Casselman, [http://www.ams.org/featurecolumn/archive/henon.html Simple Chaos-The Hénon Map]
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:*Bill Casselman, [http://www.ams.org/featurecolumn/archive/henon.html Simple Chaos-The Hénon Map]
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www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors]
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:*www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors]
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Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html Michele Henon]
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:*Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html Michele Henon]
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|ToDo=A better, less vague description of how sections of the Henon Attractor resembles the Cantor Set
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A better, less vague description of how sections of the Henon Attractor resembles the Cantor Set
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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 [http://www.ams.org/featurecolumn/archive/henon.html here].
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 [http://www.ams.org/featurecolumn/archive/henon.html here].
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Revision as of 12:02, 29 July 2009


Henon Attractor
Field: [[Field:| ]]
Image Created By: [[Author:| ]]

Henon Attractor

This image is a Henon Attractor (named after astronomer and mathematician Michel Henon), which is a fractal in the division of the chaotic strange attractor.


Contents

Basic Description

The Henon Attractor is a special kind of fractal that belongs in a group called Strange Attractors, 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. Click here to learn more about iterated functions.


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.



Magnification of the Henon Attractor
==Chaotic System==
==Changing "a" and "b"==
Here are some more examples of Henon Attractors with different ''a'' and ''b'' values.

a = 1; b = 0.542

a = 1.2; b = 0.3

a = 1.3; b = 0.3 (points need to be enlarged)

a = 1.4; b = 0.3, original Henon Attractor

a = 1.5; b = 0.2

a = 1.4; b = 0.1






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[[Category:]]







Fixed Points

Original Henon Attractor with fixed points 1 and 2
Original Henon Attractor with fixed points 1 and 2

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:


x_{n+1} = x_n\, and y_{n+1} = y_n\,


where x_n is the x-value at the nth iteration and x_{n+1} 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:

x_{n+1} = y_n + 1 - ax^2_n
y_{n+1} = bx_n\,


Since x_{n+1} = x_n\, and y_{n+1} = y_n\,, we can simplify the equations and refer to the variables as just x and y, respectively

x = y + 1 - ax^2
y = bx\,


By substituting the value of y defined by the second equation into the y in the first equation, we get

x = bx + 1 - ax^2


Using the quadratic equation

x = \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}


x = \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}


Using a = 1.4, b = 0.3:

x = 0.6314, -1.1314 \,


Using y = bx:

y = 0.1894, -0.3394 \,


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 attractor itself 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.


|other=Algebra |AuthorName=Piecewise Affine Dynamics |AuthorDesc=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". |SiteName=Lozi Maps |SiteURL=http://padyn.wikidot.com/lozi-maps |Field=Dynamic Systems |Field2=Fractals |References=:*Glenn Elert, The Chaos Hypertextbook Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and fractals

|ToDo=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. |InProgress=No |HideMME=No }}

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