Two Dimensional Maps - Instructions


a,b,c Parameters of the map functions
x_min, y_min, x_max, y_max Range of variables
transient Number of iterations to eliminate transients before plotting
Points Number of iterations plotted each plot update
Mark Size of plot mark (0=point)
Div Number of subdivisions in forming boxes for dimension calculation.
q The dimension Dq is calculated.


There are 4 different choices for plots For any choice the dimension of the plot can be estimated, although this is probably most relevant for the 2d plots.


Dimension Calculation

After the map has been iterated for some number of iterations, the generalized dimension Dq can be calculated using the box counting algorithm:
  1. Hitting Dimension will set the box counting algorithm running. The algorithm proceeds by assigning each point to a box at the finest subdivision level. A list of the occupied boxes is made, with the boxes indexed according to their position in the grid. This list is then sorted, and duplicate indices are eliminated and a counter associated with the multiply occupied box is incremented. At the next coarser level of the box counting the larger boxes are indexed, and the finer boxes are assigned to the appropriate larger box. The new list is again sorted and weeded. This process may be quite time consuming for large data sets: hit the Stop button to abort the procedure and try again with a smaller Div or number of data points.
  2. After the calculation is finished the results at each box size can be reviewed by successively hitting Continue. You might have to wait for anything to appear to happen, since the screen redraw can take a considerable time with a large number of points. If you turn off Show Boxes there is less delay.
  3. Finally hitting Plot it will plot the curve from which the dimension can be found by fitting a straight line over some portion and measuring the slope. The region fitted is chosen by clicking near two points that define the range. This process can be repeated to study how the estimate of the dimension depends on the range of points fitted.

Last modified Thursday, December 9, 1999
Michael Cross