# Two Dimensional Maps - Instructions

## Parameters

 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.

## Diagnostics

There are 4 different choices for plots
• 2D: plots the 2D map;
• L.E.: as well as plotting the 2D map, calculates the Lyapunov exponents (since this involves following the dynamics of two vectors in the tangent space, it can be slower than simply plotting the map);
• 1D-X: plots the return map Xn+1 v. Xn;
• 1D-Y: plots the return map Yn+1 v. Yn;
For any choice the dimension of the plot can be estimated, although this is probably most relevant for the 2d plots.

## Remarks

• If you change a text entry, you must hit Enter or Return on the keyboard or Restart within the applet for the change to take effect.
• The slider changes the speed of the iteration. Setting the speed too high may make the applet unresponsive or crash, depending on the Java implementation. However you will usually want to increase the speed from the default.
• The map parameters can be updated using the arrow button: the selected variable is updated by the amount in the box below the arrows.
• Dragging the mouse on a stopped plot will reset the scale to the outlined region, allowing finer detail to be seen. A single click outside the plot box will restore the default range.

## 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.

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