# Henon Attractor

(Difference between revisions)
 Revision as of 09:59, 2 July 2009 (edit)← Previous diff Revision as of 10:15, 2 July 2009 (edit) (undo)Next diff → Line 29: Line 29: ==Chaotic System== ==Chaotic System== - [[Image:Henon2.jpg|thumb|200px|left|Original Henon Attractor , a = 1.4, b = 0.3]] + {{Switch|link1=Show|link2=Hide + |1=[[Image:Henon2.jpg|thumb|200px|left]] + |2=[[Image:Henon2.jpg|thumb|200px|left|Original Henon Attractor , a = 1.4, b = 0.3]] The Henon system can be described as [[Chaos|chaotic]] and random. However, the system does have a 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 system can be described as [[Chaos|chaotic]] and random. However, the system does have a 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: Line 39: Line 41: - The Henon Attractor uses the values a = 1.4 and b = 0.3, and begin with a starting point (1,1).. + The Henon Attractor uses the values a = 1.4 and b = 0.3, and begin with a starting point (1,1). + }} ==Fractal== ==Fractal== - [[Image:HenonZoomAnimation.gif|right]] + {{Switch|link1=Show|link2=Hide + |1=[[Image:HenonZoomAnimation.gif|left]] + |2=[[Image:HenonZoomAnimation.gif|right|thumb|Zooming in on the Henon Attractor]] The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in another direction. A [[Cantor Set]] can be simply described as a iterated function beginning as a line segment that is divided into three segments, whereupon the middle segment is removed and the end two lines become the next line segments for the iterated method. The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in another direction. A [[Cantor Set]] can be simply described as a iterated function beginning as a line segment that is divided into three segments, whereupon the middle segment is removed and the end two lines become the next line segments for the iterated method. - The [[Fractal Dimension]] of the Henon Attractor is not calculable using a single equation , but it is estimated to be about 1.261. + The [[Fractal Dimension]] of the Henon Attractor is not calculable using a single equation, but it is estimated to be about 1.261. + }} ==Fixed Points== ==Fixed Points== - [[Image:HenonFixedPoints1.png|thumb|300px|right|Original Henon Attractor with fixed points]] + {{Switch|link1=Show|link2=Hide - + |1=[[Image:HenonFixedPoints1.png|300px|left]] + |2=[[Image:HenonFixedPoints1.png|thumb|300px|right|Original Henon Attractor with fixed points]] As seen from the previous system of equations, the Henon Attractor uses only two variables (x and y) that are evaluated into themselves, which results in two equilibrium or fixed points for the attractor. This points are very unique, because if the system ever plotted onto the fixed points, the plotted values would no longer change and would remain at the fixed points. As seen from the previous system of equations, the Henon Attractor uses only two variables (x and y) that are evaluated into themselves, which results in two equilibrium or fixed points for the attractor. This points are very unique, because if the system ever plotted onto the fixed points, the plotted values would no longer change and would remain at the fixed points. Line 62: Line 69: {{HideThis|1=Solving the System of Equations|2= {{HideThis|1=Solving the System of Equations|2= - $x_{n+1} = y_n + 1 - ax^2_n$ becomes $x = y + 1 - ax^2$ + $x_{n+1} = y_n + 1 - ax^2_n$ - $y_{n+1} = bx_n\,$ becomes $y = bx\,$ + $y_{n+1} = bx_n\,$ - $x = bx + 1 - ax^2$, using the quadratic equation + If $x_{n+1} = x\,$ and $y_{n+1} = y\,$ then - $x_{1,2} = \frac{-b \pm sqrt{b^2 - 4ac}}{2a}$ + - $x_{1,2} = \frac{-(b-1) \pm sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}$ + :$x = y + 1 - ax^2$ - $x_{1,2} = \frac{-(b-1) \pm sqrt{(b-1)^2 + 4a}}{-2a)}$ + :$y = bx\,$ + + + :$x = bx + 1 - ax^2$ + + + Using the quadratic equation + + + :$x_{1,2} = \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}$ + + + :$x_{1,2} = \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}$ + + + :$x_{1,2} = 0.6314,-1.1314 \,$ - $x_{1,2} = 0.6314,-1.1314 \,$ Using $y = bx\,$: Using $y = bx\,$: - $y_{1,2} = 0.1894,-0.3394 \,$ + :$y_{1,2} = 0.1894,-0.3394 \,$ }} }} There are two types of fixed points, '''stable''' and '''unstable'''. The first fixed point (0.6314, 0.1894), that is labeled "1" on the image to the right 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. There are two types of fixed points, '''stable''' and '''unstable'''. The first fixed point (0.6314, 0.1894), that is labeled "1" on the image to the right 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. - + }} |other=Algebra |other=Algebra |AuthorName=SiMet |AuthorName=SiMet Line 101: Line 121: 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] |InProgress=Yes |InProgress=Yes + |HideMME=No }} }}

## Revision as of 10:15, 2 July 2009

Henon Attractor
This image is a variation of the Henon Attractor, which is a fractal in the division of the chaotic Strange Attractors named after astronomer Michel Henon. The Henon Attractor emerged from Henon's attempt to model the orbits of celestial objects.

# Basic Description

The Henon Attractor is a special kind of fractal that belongs in a group called Strange Attractors, a category of images that is very strange indeed.

A characteristic of this strange function is that it drawn irregularly. If you iterate the functions that describe the Henon Attractor and plot the points of the functions for each iteration in a time sequence, you would observe that the points jump from one random location within the image to another. If you take a look at the animation, you can see the irregularity of the first 120,000 plotted points. Eventually, the individual points become so numerous that they appear to form lines and an image emerges.

Also, this image is 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 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

This image is a modified version of the famous Henon Attractor that will be described below.

## Chaotic System

Original Henon Attractor , a = 1.4, b = 0.3

The Henon system can be described as chaotic and random. However, the system does have a 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:

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

$y_{n+1} = bx_n\,$

The Henon Attractor uses the values a = 1.4 and b = 0.3, and begin with a starting point (1,1).

## Fractal

Zooming in on the Henon Attractor

The shape of the Henon Attractor is often described as a smooth fractal in one direction and as a Cantor Set in another direction. A Cantor Set can be simply described as a iterated function beginning as a line segment that is divided into three segments, whereupon the middle segment is removed and the end two lines become the next line segments for the iterated method.

The Fractal Dimension of the Henon Attractor is not calculable using a single equation, but it is estimated to be about 1.261.

## Fixed Points

Original Henon Attractor with fixed points

As seen from the previous system of equations, the Henon Attractor uses only two variables (x and y) that are evaluated into themselves, which results in two equilibrium or fixed points for the attractor. This points are very unique, because if the system ever plotted onto the fixed points, the plotted values would no longer change and would remain at the fixed points.

The two fixed points of the Henon Attractor must satisfy $x_{n+1} = x\,$ and $y_{n+1} = y\,$.

Using the Henon Attractor's system of equations, the fixed points are (0.6314 , 0.1894) and (-1.1314 , -0.3394).

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

$y_{n+1} = bx_n\,$

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

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

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

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

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), that is labeled "1" on the image to the right 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 Relates

The artistic image of the Henon Attractor that is the subject of this page instead uses the values a = 1 and b = 0.542.

# 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

Have questions about the image or the explanations on this page?