There is yet another way to generate striking graphs.  In fact these graphs can be quite spectacular.  

To graph iterative equations on the computer, enter the following procedure into some computer system:

 

             Enter starting x and y values

                        Use two equations to compute new x and y: plot them

                        Use the new x and y as inputs to the same equations; plot

             Repeat say 100 times

 

                        Now repeat again for a new starting x and y.

Iterative equations use a new algorithm. Make a rule that changes one point into another point. Often the best way to make a rule is with an equation.  Since there are two coordinates, there will be two equations to tell how to find the new x and y coordinates (called newx and newy) in terms of x and y, the old coordinates. For example, these equations are known to make interesting patterns.  

 

        newy = a - x  (start with a = .2)

 

A procedure that is repeated over and over, with the previous result always being the new starting point, is called an iterative procedure. The graph that is traced out is called the “orbit” of the starting point. In this case the orbit is somewhat circular.

 

Iterative equations are different from the graphs of the functions we have been studying.  They have a much wider variety of behavior. In particular numbers can get very large very fast. Here are some tips for using this program to graph iterative equations.

 

How to explore iterative equations

Start with the pair of equations above. This equation does not like large numbers (because of the square) so the interesting behavior is in the region between -1 and 1. This means that (.3,.3) is an interesting starting point, while (3,3) is not. To get a better view, zoom in by choosing a larger scale (50 is a good choice). One starting point produces one orbit, such as shown at the right. Explore more orbits by choosing more starting points.  

 

Change the value of a

The number .5 in the original equation can be changed. It is usually called the parameter a. After you have explored the first equation, try another by changing the value for a. This will give you patterns such as those on the next page. See if you can find the little periodic orbits.

 

After playing with the sin(x) function, try some other functions. Zoom in or out as appropriate. You will find some nice patterns.

 

Change both equations

Try anything. Keep xnew and ynew as the left side of each equation, but try anything you want with x's and y's on the right. Zoom in or out as appropriate. You should find some attractors and repellors by doing this (see below and the opposite page for the vocabulary of iterative equations).