Iterating functions with Complex Numbers and the Mandelbrot Set

Iteration is the process of repeating a task over and over again.  In reference to functions, the process of iterating a function refers to obtaining a value from a function, using that value to produce a subsequent value, using that subsequent value to produce yet another value, and so on.

Example of function iteration:
f(x) = x2 + 2 ; starting value of  x = 0.

           f(0) = 2
          f(2) = 6
          f(6) = 38
          f(38) = 1446
          and so on.....

 Example of iteration with a complex number:
f(x) = 2x + (4 - 2i)  ; starting value of x = 0.

 f(0) = 4 - 2i
 f(4-2i) = 2(4-2i) + (4-2i) = 12 - 6i
 f(12-6i) = 2(12-6i) + (4-2i) = 28 - 14i
 f(28-14i) = 2(28-14i) + (4-2i) = 60 - 30i
 and so on ......

This process of iteration can serve as a good exercise for students to practice their basic skills for manipulating complex numbers.  Have students consider iterations on the following functions (or create your own):

f(x) = x2 + (1 - 3i) ; starting value of x = 0;  four iterations

f(x) = 3x - (2 - i) ; starting value of x = 0; three iterations

f(x) = 2x2 + (1 + i); starting value of x = 0; three iterations

 

Graphically speaking, a function of the form f(x) = x2 + c (where c is a complex number) is a special function under iteration.  If you plot the results of the iterations (starting at x = 0) in the complex plane, you obtain what is called the critical orbits of c.  If these critical orbits repeat (where the same point in the complex plane repeats), the complex number is in the Mandelbrot set.  If the critical orbits simply move further and further away from the origin, the complex numbers are not in the Mandelbrot set.


 

Examine the following functions to see where the orbits fall in the Mandelbrot set:

     f(x) = x2 + (-1.1 + 0.2i)

     f(x) = x2 + (-0.05 + 0.75i)

Use the graphing calculator
to see if critical orbits repeat and lie in the Mandelbrot set:
(instructions for the TI-83+)

f(x) = x2 + (- 0.6 + 0.2i)
The screens below show the iterations of the function and the start of a repeating nature of the orbits.  Put calculator in a+bi mode.

 

Further Investigations:

Examine the critical orbits of any complex number and the critical orbits of its complex conjugate.  A discussion of the symmetric nature of the Mandelbrot set should follow this examination..

 

 


Roberts