Calculus 231 Project 1
Chaos Theory
Composites of functions play an important role in the subject of chaos, which, in addition to mathematics, has attracted much attention in such diverse fields as meteorology and medicine. Consider a function f whose range is contained in its domain. Then the numbers

f(x), f(f(x)), f(f(f(x))), ...

are called iterates of f at x.
Consider f(x) = x2 + c
Enter a value for c:
Enter the function:
First iterate:
Second iterate:
The Madelbrot Set
Third iterate:
For our purposes, it will be more desirable to enter the function in subscripted notation
Number of iterates:
Initial value:
Recursive values:
Table of values:
Notice the values match those obtained with conventional notation
For some values of c, the function will increase without bound. For some values of c, the function will approach a limit. For some values of c, the function will oscillate. You are to determine the intervals of c for which the function exhibits these types of behavior. Although it is possible to obtain these intervals analytically, the necessary methods are beyond the scope of this course. You are to use numerical and graphical evidence in your investigation. You must present at least two examples of each case. There should be two intervals for which the function increases without bound, one where the function reaches a limit from one side, one where it approaches a limit in an oscillating fashion, and one where the function oscillates without approaching a limit.
For additional information on Chaos Theory, visit one of the following sites
http://www.students.uiuc.edu/~ag-ho/chaos/
http://www.susqu.edu/facstaff/b/brakke/complexity/hagey/chaos.htm
http://www-chaos.umd.edu/
Sample investigation
Enter a value for c:
Number of iterates:
Enter the initial value:
Enter the function:
Graphical evidence:
Notice that the graph increases without bound
Numerical evidence:
Notice that the numbers increase without bound
Sample investigation
Enter a value for c:
Number of iterates:
Enter the initial value:
Enter the function:
Graphical evidence:
Notice that the graph approaches a limit
Numerical evidence:
Notice that the numbers approach 0.49