# Edit Edit an Image Page: Henon Attractor

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 Image Title*: Upload a Math Image 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. 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=== [[Image:HenonAnimation.gif|thumb|250px|left]] Say we took a single starting point (x,y) and plotted it on a graph. Then, we applied the two Henon Attractor equations to the initial point and emerged with a new point that we graphed. Next, we took this new point and again applied the two equations to it and graphed the next new point. If we continued to apply the two equations to each new point in a process called '''iteration''' and plotted every outcome from this iteration, we would create a Henon Attractor. Click here to learn more about [[Iterated Functions|iterated functions]]. Furthermore, if we plotted each outcome one at a time, we would observe that the points jump from one random location to another within the image. If you take a look at the animation, you can see the irregularity of the plotted points. Eventually, the individual points become so numerous that they appear to form lines and an image emerges. ===Magnification of the Henon Attractor=== Image:HenonMag1.png ==Fractal Properties== The Henon Attractor is often described as being similar to the [[Cantor Set]]. Let us zoom into the Henon Attractor near the doubled-tip of the fractal (as seen in the animation). We can see that as we continue to magnify the lines that form the structure of the Henon Attractor, these lines become layers of increasingly deteriorating lines that appear to resemble the Canter Set. The [[Fractal Dimension]] of the Henon Attractor is not calculable using a single equation, but it is estimated to be about 1.261. ==Chaotic System== [[Image:Henon2.jpg|right|thumb|Original Henon Attractor, a = 1.4, b = 0.3]] The Henon system can be described as [[Chaos|chaotic]] and random. However, the system does have structure in that its points settle very close to an underlying pattern called a chaotic attractor. 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. Image:Henon.jpg 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". Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other None Algebra Analysis Calculus Dynamic Systems Fractals Geometry Graph Theory Number Theory Polyhedra Topology Other :*Glenn Elert, [http://hypertextbook.com/chaos/21.shtml The Chaos Hypertextbook] :*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] :*Michele Henon, [http://www.exploratorium.edu/turbulent/CompLexicon/henon.html Michele Henon] 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]. Yes, it is.