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Henon Attractor - Math Images

Henon Attractor

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|ImageIntro=This image is a Henon Attractor (named after astronomer Michel Hénonn), which is a fractal in the division of the chaotic strange attractor. The Henon Attractor emerged from Hénon's attempt to model the orbits of celestial objects.
|ImageIntro=This image is a Henon Attractor (named after astronomer Michel Hénonn), which is a fractal in the division of the chaotic strange attractor. The Henon Attractor emerged from Hénon's attempt to model the orbits of celestial objects.
|ImageDescElem=
|ImageDescElem=
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The 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.
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===Making the Henon Attractor===
<div style="width:265px; height:225px; float:left;">{{#eqt: HenonAnimation.avi|250|left}}</div>
<div style="width:265px; height:225px; float:left;">{{#eqt: HenonAnimation.avi|250|left}}</div>
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The Henon Attractor is a special kind of fractal that belongs in a group called [[Strange Attractors]].
 
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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.
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Say we took a single starting point (x,y) and plotted it on a graph. Then, we applied the <balloon title="One equation for the x-value and one equation for the y-value.">two</balloon> 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.
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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 a number of plotted points. Eventually, the individual points become so numerous that they appear to form lines and an image emerges.
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This image results from an [[Iterated Functions|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.
 
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Image:HenonMag4.png|512X
Image:HenonMag4.png|512X
</gallery>
</gallery>
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This image results from an [[Iterated Functions|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.
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|Pre-K=No
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{{hide|1=
{{hide|1=
[[Image:Henon2.jpg|right|thumb|Original Henon Attractor, a = 1.4, b = 0.3]]
[[Image:Henon2.jpg|right|thumb|Original Henon Attractor, a = 1.4, b = 0.3]]
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The Henon system can be described as [[Chaos|chaotic]] and random. However, the system does have structure in that its points settle very close to an underlying pattern called a <balloon title="In this chaotic system, points are in fact over time evolving towards a recognizable structure that is called the chaotic attractor."> chaotic attractor></balloon> or <balloon title="" basin of attraction></balloon>. The basic Henon Attractor can be described by the equations:
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The Henon system can be described as [[Chaos|chaotic]] and random. However, the system does have structure in that its points settle very close to an underlying pattern called a <balloon title="In a chaotic system like this fractal, over time plotted points are evolving towards (or being "attracted" to) a region that is called the chaotic attractor.">chaotic attractor</balloon>. The basic Henon Attractor can be described by the equations, where <math>x_n</math> is the x-value at the nth iteration.
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{{Switch|link1=Show|link2=Hide
{{Switch|link1=Show|link2=Hide
|1=[[Image:HenonFixedPoints1.png|180px]]
|1=[[Image:HenonFixedPoints1.png|180px]]
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|2=[[Image:HenonFixedPoints1.png|thumb|300px|right|Original Henon Attractor with fixed points]]
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|2=[[Image:HenonFixedPoints1.png|thumb|325px|right|Original Henon Attractor with fixed points 1 and 2]]
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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.
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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 or in algebraic terms:
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:<math>x_{n+1} = x\,</math> and <math>y_{n+1} = y\,</math>
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:where <math>x_n</math> is the x-value at the nth iteration and <math>x_{n+1}</math> is the x-value at the next iteration.
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The two fixed points of the Henon Attractor must satisfy <math>x_{n+1} = x\,</math> and <math>y_{n+1} = y\,</math>.
 
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Using the Henon Attractor's system of equations, the fixed points are (0.6314 , 0.1894) and (-1.1314 , -0.3394).
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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, the fixed points for the original Henon Attractor are (0.6314 , 0.1894) and (-1.1314 , -0.3394).
{{HideThis|1=Solving the System of Equations|2=
{{HideThis|1=Solving the System of Equations|2=
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www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors]
www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors]
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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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|ToDo=
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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].
|InProgress=Yes
|InProgress=Yes
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Revision as of 15:31, 8 July 2009

Image:inprogress.png
Henon Attractor
This image is a Henon Attractor (named after astronomer Michel Hénonn), which is a fractal in the division of the chaotic strange attractor. The Henon Attractor emerged from Hénon's attempt to model the orbits of celestial objects.
Henon Attractor
Fields: Dynamic Systems and Fractals
Created By: SiMet

Contents

Basic Description

The 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 a number of plotted points. Eventually, the individual points become so numerous that they appear to form lines and an image emerges.



Magnification of the Henon Attractor

1X

8X

64X

512X


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

Fractal Properties

Zooming in on the Henon Attractor
Zooming in on the Henon Attractor


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.



Chaotic System

Original Henon Attractor, a = 1.4, b = 0.3
Original Henon Attractor, a = 1.4, b = 0.3

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. The basic Henon Attractor can be described by the equations, where x_n is the x-value at the nth iteration.


x_{n+1} = y_n + 1 - ax^2_n


y_{n+1} = bx_n\,


The original Henon Attractor (the featured image at the top of the page) created by astronomer Michel Hénon uses the values a = 1.4 and b = 0.3 and begins with a starting point (1,1). However, by changing the values of a and b, we can obtain Henon Attractors that look slightly different.


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 or in algebraic terms:


x_{n+1} = x\, and y_{n+1} = y\,


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, 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\,


If x_{n+1} = x\, and y_{n+1} = y\, then

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


x = bx + 1 - ax^2


Using the quadratic equation

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


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


Using a = 1.4, b = 0.3:

x_{1,2} = 0.6314,-1.1314 \,


Using y = bx:

y_{1,2} = 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 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.


Changing "a" and "b"

Although the original Henon Attractor uses the values a = 1.4 and b = 0.3, we can alter those values within a range to produce various-looking Henon Attractors. As we can see below, a and b are limited to the range of values outside of which the fractal ceases to resemble the Henon Attractor.


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

a = 1.5; b = 0.2

a = 1.4; b = 0.1




Teaching Materials

There are currently no teaching materials for this page. Add teaching materials.

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".



References

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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