# Henon Attractor

### From Math Images

(Difference between revisions)
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|----------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|------------------------------------------------------|----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Current revision (10:22, 28 June 2012) (edit) (undo) |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| (42 intermediate revisions not shown.) |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| Line 1:                                |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Line 1:                                              |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | {{Image Description                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | {{Image Description Ready                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        | |ImageName=Henon Attractor                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                      | |ImageName=Henon Attractor                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
| -                                      | |Image=Henon.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    | |Image=HenonMain.jpg                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      | |ImageIntro=This image is a variation of the Henon Attractor (named after astronomer Michel Henon), which is a fractal in the division of the chaotic strange attractor. The Henon Attractor emerged from Henon'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.                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <div style="width:265px; height:225px; float:left;">{{#eqt: HenonAnimation.avi|250|left}}</div>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | 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 <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. 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.                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <gallery caption="Magnification of the Henon Attractor" widths="220px" heights="150px" perrow="4">                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    | ===Magnification of the Henon Attractor===                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | <gallery caption="" widths="220px" heights="150px" perrow="4">                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
|                                        | 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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
|                                        | </gallery>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |                                                      | </gallery>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | 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 26:                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Line 36:                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |MiddleSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |                                                      | |MiddleSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        |
|                                        | |HighSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      | |HighSchool=Yes                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |ImageDesc= The image featured at the top of this page is a modified version of the Henon Attractor that will be described below.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        | +                                                    | |ImageDesc=                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | ==Fractal Properties==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                      | ==Fractal Properties==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
| -                                      | |1=[[Image:HenonZoomAnimation.gif|150px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |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. If we zoom into the Henon Attractor near the doubled-tip of the fractal (as seen in the animation), we can see that the points form layers of lines that appear to resemble the Canter Set. If we follow the attractor backwards from the doubled-tip, we can see that the fractal is more smooth and contains less bundles of lines.                                                                                                                                                                                             | +                                                    | 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 <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.                                                                                                                                                                                                                                                                                                                                                                                     |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | ==Chaotic System==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                      | ==Chaotic System==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
| -                                      | |1=[[Image:Henon2.jpg|150px]]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |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 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:                                                                                                                                                                                                                                                                                                                                                                 |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | [[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 <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.                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>x_{n+1} = y_n + 1 - ax^2_n</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  | +                                                    | ::<math>x_{n+1} = y_n + 1 - ax^2_n</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | ::<math>y_{n+1} = bx_n\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | <math>y_{n+1} = bx_n\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            | +                                                    | 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 Henon Attractor uses the values a = 1.4 and b = 0.3 and begin with a starting point (1,1).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                    | ==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.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | <gallery widths="250px" heights="250px" perrow="3" align ="center">                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | 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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | </gallery>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | ==Fixed Points==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                      | ==Fixed Points==                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | {{Switch|link1=Show|link2=Hide                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           | +                                                    | Looking at the <balloon title="load:eqns">system of equations</balloon> <span id="eqns" style="display:none"><math>x_{n+1} = y_n + 1 - ax^2_n</math> and <math>y_{n+1} = bx_n\,</math></span> 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 <math>x_{n+1} = x\,</math>  and  <math>y_{n+1} = y\,</math>.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    | +                                                    | :<math>x_{n+1} = x_n\,</math>  and  <math>y_{n+1} = y_n\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | Using the Henon Attractor's system of equations, the fixed points are (0.6314 , 0.1894) and (-1.1314 , -0.3394).                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | :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.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | 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 <math>x_{n+1} = x\,</math>  and  <math>y_{n+1} = y\,</math> then                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | Since <math>x_{n+1} = x_n\,</math>  and  <math>y_{n+1} = y_n\,</math>, we can simplify the equations and refer to the variables as just <math>x</math> and <math>y</math>, respectively                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | :<math>x = y + 1 - ax^2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |                                                      | :<math>x = y + 1 - ax^2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           |
| Line 81:                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Line 97:                                             |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | :<math>y = bx\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                      | :<math>y = bx\,</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | By substituting the value of <math>y</math> defined by the second equation into the <math>y</math> in the first equation, we get                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | :<math>x = bx + 1 - ax^2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      | :<math>x = bx + 1 - ax^2</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| Line 86:                               |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | Line 104:                                            |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | Using the <balloon title = "load:quadeqn"> quadratic equation </balloon>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |                                                      | Using the <balloon title = "load:quadeqn"> quadratic equation </balloon>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                 |
| -                                      | <span id="quadeqn" style="display:none"> <math>x_{1,2} = \frac{-b \pm sqrt{b^2 - 4ac}}{2a}</math> </span>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | +                                                    | <span id="quadeqn" style="display:none"> <math>x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}</math> </span>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>x_{1,2} =  \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              | +                                                    | :<math>x =  \frac{-(b-1) \pm \sqrt{(b-1)^2 - 4(-a)(1)}}{2(-a)}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>x_{1,2} =  \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                      | +                                                    | :<math>x =  \frac{-(b-1) \pm \sqrt{(b-1)^2 + 4a}}{-2a}</math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | Using a = 1.4, b = 0.3:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |                                                      | Using a = 1.4, b = 0.3:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>x_{1,2} = 0.6314,-1.1314 \, </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | :<math>x = 0.6314, -1.1314 \, </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | Using y = bx:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |                                                      | Using y = bx:                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                            |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | :<math>y_{1,2} = 0.1894,-0.3394 \, </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               | +                                                    | :<math>y = 0.1894, -0.3394 \, </math>                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                    |
|                                        | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |                                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
|                                        |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          | +                                                    | 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.                                                                                                                                                                                                         | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | }}                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       | +                                                    |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |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                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
| -                                      | |ImageRelates= The artistic image of the Henon Attractor that is the subject of this page instead uses the values a = 1 and b = 0.542.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                   |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
|                                        | |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]                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
| -                                      | |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

1X  8X  64X  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.

# A More Mathematical Explanation

Note: understanding of this explanation requires: *Algebra
[Click to view A More Mathematical Explanation]

## Fractal Properties

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

[Click to hide A More Mathematical Explanation]

## 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$D = \frac{log(n)}{log(e)}$, 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.

|                                            |                  |                                               |
|--------------------------------------------|------------------|-----------------------------------------------|
| 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, original Henon Attractor | a = 1.5; b = 0.2 | a = 1.4; b = 0.1                              |

## Fixed Points

Looking at the system of equations $x_{n+1} = y_n + 1 - ax^2_n$ and $y_{n+1} = bx_n\,$ 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).

[Show Solving the System of Equations][Hide Solving the System of Equations]

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

$x = bx + 1 - ax^2$

Using the quadratic equation $x_{1,2} = \frac{-b \pm \sqrt{b^2 - 4ac} }{2a}$

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

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

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.

# 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.

Leave a message on the discussion page by clicking the 'discussion' tab at the top of this image page.
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 26, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": false, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8760894536972046, "perplexity_flag": "middle"}