Design your own Web Page Background

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Directions:
  1. Scroll Bar Image Set the parameters for an area preserving map on the torus by clicking on the eight scroll bars in the upper right hand courner of the applet above. The image will look best if you modify the values of each of the eight scroll bars. 
  2. Choose a color
    • Click somewhere on on the circle to choose the Hue and Saturation of a color.
    • Click on the bar below to modify the brightness of a color.
    • The next bar down shows the color you have selected. You can click on this bar at any time to replace make this color the background color for your image (currently it is black).
    • If you intend to use the image you create as a web page background, be careful to choose aesthetically appealing colors.
  3. After choosing a color different from the background color, you may click anywhere in the large rectangle in the upper left of the applet to plot an orbit of your chosen area preserving map.
  4. Continue choosing colors and plotting orbits until you are happy with the picture you have created. You can choose new values for the scroll bars as in step 1, but this will reset the image you have created to only the background color.
  5. Submit Button After you are satified with the picture you have created, click the submit button to view and download your image.

The Mathematics:
  • You are plotting the orbit of points under a return map of the torus that preserves area, specifically this map is called a Generalized Standard Map. The map is determined as follows:
    • Choose two periodic functions:  f,g : R/Z --> R/Z
    • Given any x,y define
      • y' = y + f(x)
      • x' = x + g(y')
    • The map is  (x,y) |--> (x',y')
  • These maps tend to have large regions of chaos, with "islands" of order around fixed points. Nearby each of these islands there are smaller islands containing periodic orbits. Each of these islands then has a family of neigboring islands as well, and so on. This is typical of area preserving maps.
  • A reference on the subject is RS MacKay, Renormalization in area preserving maps
View the Source Code of this applet