Henon Attractor
From Math Images
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Image=HenonMain.jpg  Image=HenonMain.jpg  
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.  
  ImageDescElem=  +  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. 
  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===  
<gallery caption="" widths="220px" heights="150px" perrow="4">  <gallery caption="" widths="220px" heights="150px" perrow="4">  
  Image:HenonMag1.png  +  Image:HenonMag1.png 
  +  ImageDesc===Fractal Properties==  
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  ==Fractal Properties==  +  
{{Switchlink1=Showlink2=Hide  {{Switchlink1=Showlink2=Hide  
1=[[Image:HenonZoomAnimation.gif150px]]  1=[[Image:HenonZoomAnimation.gif150px]]  
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The [[Fractal Dimension]] of the Henon Attractor is not calculable using a single <balloon title = "load:equation"> equation</balloon><span id="equation" style="display:none"><math>D = \frac{log(n)}{log(e)}</math></span>, but it is estimated to be about 1.261.  The [[Fractal Dimension]] of the Henon Attractor is not calculable using a single <balloon title = "load:equation"> equation</balloon><span id="equation" style="display:none"><math>D = \frac{log(n)}{log(e)}</math></span>, but it is estimated to be about 1.261.  
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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  other=Algebra  
AuthorName=Piecewise Affine Dynamics  AuthorName=Piecewise Affine Dynamics  
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Field=Dynamic Systems  Field=Dynamic Systems  
Field2=Fractals  Field2=Fractals  
  +  References=Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook]  
  References=  +  
  Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook]  +  
HeinzOtto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals'''  HeinzOtto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals'''  
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Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html Michele Henon]  Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html Michele Henon]  
  ToDo=  +  ToDo=A better, less vague description of how sections of the Henon Attractor resembles the Cantor Set 
  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 [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].  
+  InProgress=No  
HideMME=No  HideMME=No  
}}  }} 
Revision as of 14:05, 17 July 2009
Henon Attractor 

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.


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[[Category:]]
Fixed Points
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:
 and
 where is the xvalue at the nth iteration and is the xvalue 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:
Since and , we can simplify the equations and refer to the variables as just and , respectively
By substituting the value of defined by the second equation into the in the first equation, we get
Using the quadratic equation
Using a = 1.4, b = 0.3:
Using y = bx:
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/lozimaps
Field=Dynamic Systems
Field2=Fractals
References=Glenn Elert, The Chaos Hypertextbook
HeinzOtto Peitgen, Hartmut Jürgens, Dietmar Saupe, Chaos and fractals
Bill Casselman, Simple ChaosThe Hénon Map
www.ibiblio.org Henon Strange Attractors
Michele Henon, Michele Henon 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 }}