# Henon Attractor

(Difference between revisions)
 Revision as of 11:28, 8 July 2009 (edit)← Previous diff Current revision (09:22, 28 June 2012) (edit) (undo) (29 intermediate revisions not shown.) Line 1: Line 1: - {{Image Description + {{Image Description Ready |ImageName=Henon Attractor |ImageName=Henon Attractor |Image=HenonMain.jpg |Image=HenonMain.jpg - |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 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. -
{{#eqt: HenonAnimation.avi|250|left}}
- The Henon Attractor is a special kind of fractal that belongs in a group called [[Strange Attractors]]. + ===Making the Henon Attractor=== + [[Image:HenonAnimation.gif|thumb|250px|left]] - 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. + 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|iterated functions]]. - 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. + 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=== + Image:HenonMag1.png|1X Image:HenonMag1.png|1X Image:HenonMag2.png|8X Image:HenonMag2.png|8X Line 21: Line 23: Image:HenonMag4.png|512X Image:HenonMag4.png|512X + + + 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. + + + ===History of the Henon Attractor=== + 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. + |Pre-K=No |Pre-K=No Line 28: Line 38: |ImageDesc= |ImageDesc= ==Fractal Properties== ==Fractal Properties== - {{Switch|link1=Show|link2=Hide - |1=[[Image:HenonZoomAnimation.gif|150px]] - |2=[[Image:HenonZoomAnimation.gif|right|thumb|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 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. 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== ==Chaotic System== - {{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]] - 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 '''chaotic attractor''' or '''basin of attraction'''. The basic Henon Attractor can be described by the equations: + 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 chaotic attractor. The basic Henon Attractor can be described by the equations, where $x_n$ is the x-value at the nth iteration. - + ::$x_{n+1} = y_n + 1 - ax^2_n$ ::$x_{n+1} = y_n + 1 - ax^2_n$ - ::$y_{n+1} = bx_n\,$ ::$y_{n+1} = bx_n\,$ + Astronomer Michel Henon created the original Henon Attractor using the values ''a'' = 1.4 and ''b'' = 0.3 and starting point (1,1). These are also the values used by the artist to create the featured image at the top of the page. However, by changing the values of ''a'' and ''b'', we can obtain Henon Attractors that look slightly different. - 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. + - }} + ==Changing "a" and "b"== + Although the original Henon Attractor uses ''a'' = 1.4 and ''b'' = 0.3, we can alter those values slightly to produce various-looking Henon Attractors. However, the values of ''a'' and ''b'' are limited to a small 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. + + + Image:Henon.jpg|a = 1; b = 0.542 + Image:Henon_a1.2_b0.3.png|a = 1.2; b = 0.3 + Image:Henon_a1.3_b0.3.png|a = 1.3; b = 0.3 (points need to be enlarged) + Image:Henon_a1.4_b0.3.png|a = 1.4; b = 0.3, original Henon Attractor + Image:Henon_a1.5_b0.2.png|a = 1.5; b = 0.2 + Image:Henon_a1.4_b0.1.png|a = 1.4; b = 0.1 + ==Fixed Points== ==Fixed Points== - {{Switch|link1=Show|link2=Hide + 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: - |1=[[Image:HenonFixedPoints1.png|180px]] + - |2=[[Image:HenonFixedPoints1.png|thumb|300px|right|Original Henon Attractor with 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 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. + - The two fixed points of the Henon Attractor must satisfy $x_{n+1} = x\,$ and $y_{n+1} = y\,$. + :$x_{n+1} = x_n\,$ and $y_{n+1} = y_n\,$ - Using the Henon Attractor's system of equations, the fixed points are (0.6314 , 0.1894) and (-1.1314 , -0.3394). + + :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). {{HideThis|1=Solving the System of Equations|2= {{HideThis|1=Solving the System of Equations|2= Line 75: Line 91: - If $x_{n+1} = x\,$ and $y_{n+1} = y\,$ then + 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 :$x = y + 1 - ax^2$ :$x = y + 1 - ax^2$ Line 81: Line 97: :$y = bx\,$ :$y = bx\,$ + + By substituting the value of $y$ defined by the second equation into the $y$ in the first equation, we get :$x = bx + 1 - ax^2$ :$x = bx + 1 - ax^2$ Line 86: Line 104: Using the quadratic equation Using the quadratic equation - + - :$x_{1,2} = \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}$ + :$x = \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}$ - :$x_{1,2} = \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}$ + :$x = \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}$ Using a = 1.4, b = 0.3: Using a = 1.4, b = 0.3: - :$x_{1,2} = 0.6314,-1.1314 \,$ + :$x = 0.6314, -1.1314 \,$ Using y = bx: Using y = bx: - :$y_{1,2} = 0.1894,-0.3394 \,$ + :$y = 0.1894, -0.3394 \,$ }} }} - + 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. - 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"== + - {{Switch|link1=Show|link2=Hide|1=[[Image:Henon.jpg|200px]]|2= + - 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, the range of values that can be applied to ''a'' and ''b'' is limited to about < ''a'' < and < ''b'' < or the fractal ceases to resemble the Henon Attractor. + - + - + - Here are some more examples of Henon Attractors with different ''a'' and ''b'' values. + - + - + - Image:Henon.jpg|a = 1; b = 0.542 + - Image:Henon_a1.2_b0.3.png|a = 1.2; b = 0.3 + - Image:Henon_a1.3_b0.3.png|a = 1.3; b = 0.3 (points were enlarged) + - Image:Henon_a1.4_b0.3.png|a = 1.4; b = 0.3 + - Image:Henon_a1.5_b0.2.png|a = 1.5; b = 0.2 + - Image:Henon_a1.4_b0.1.png|a = 1.4; b = 0.1 + - + - + - + - }} + |other=Algebra |other=Algebra - |AuthorName=SiMet + |AuthorName=Piecewise Affine Dynamics - |AuthorDesc=The images created by this author were found on the author's (username SiMet) Picasa Web Album under the category "Computer Art". + |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=SiMet's Gallery (Picasa) + |SiteName=Lozi Maps - |SiteURL=http://picasaweb.google.com/lh/photo/6iojFK86zEuRY3Zi5DBq-w + |SiteURL=http://padyn.wikidot.com/lozi-maps |Field=Dynamic Systems |Field=Dynamic Systems |Field2=Fractals |Field2=Fractals |FieldLinks = |FieldLinks = |References= |References= - Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook] + :*Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook] - Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals''' + :*Heinz-Otto Peitgen, Hartmut Jürgens, Dietmar Saupe, '''Chaos and fractals''' + + :*Bill Casselman, [http://www.ams.org/featurecolumn/archive/henon.html Simple Chaos-The Hénon Map] + + :*www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors] - Bill Casselman, [http://www.ams.org/featurecolumn/archive/henon.html Simple Chaos-The Hénon Map] + :*Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html 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 [http://www.ams.org/featurecolumn/archive/henon.html here]. - www.ibiblio.org [http://www.ibiblio.org/e-notes/Chaos/strange.htm Henon Strange Attractors] - |ToDo=A better, less vague description of how sections of the Henon Attractor resembles the Cantor Set - |InProgress=Yes - |HideMME=No }} }}

## Current revision

Henon Attractor
Fields: Dynamic Systems and Fractals
Image Created By: Piecewise Affine Dynamics
Website: Lozi Maps

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.

# 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

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.

### History of the Henon Attractor

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.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra

## Fractal Properties

The Henon Attractor is often described as being similar to the Cantor Set [...]

## Fractal Properties

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

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.

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

Astronomer Michel Henon created the original Henon Attractor using the values a = 1.4 and b = 0.3 and starting point (1,1). These are also the values used by the artist to create the featured image at the top of the page. However, by changing the values of a and b, we can obtain Henon Attractors that look slightly different.

## Changing "a" and "b"

Although the original Henon Attractor uses a = 1.4 and b = 0.3, we can alter those values slightly to produce various-looking Henon Attractors. However, the values of a and b are limited to a small 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.

## 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:

$x_{n+1} = x_n\,$ 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

$x = bx + 1 - ax^2$

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

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

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.

# 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".

# References

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.