Henon Attractor

From Math Images

(Difference between revisions)
Jump to: navigation, search
Line 2: Line 2:
|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]]. 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.
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.
Line 28: Line 28:
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.
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.
 +
 +
 +
====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.
Line 60: Line 65:
-
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.
+
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.
}}
}}
Line 156: Line 161:
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]
 +
 +
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

Revision as of 15:36, 9 July 2009


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




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.


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

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\,


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 the values a = 1.4 and b = 0.3, we can alter those values slightly 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.



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\, 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, 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\,


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 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.




Teaching Materials

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

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

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

Michele Henon, Michele Henon

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.




If you are able, please consider adding to or editing this page!

Have questions about the image or the explanations on this page?
Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.






Personal tools