Even very simple dynamical systems, such as the one in this lab, can result in highly complicated behavior. Most of the time, it is impossible to give exact numerical values at which behavior occurs in a dynamical system. We can only study the qualitative behavior. Thus the qualitative approach, such as you will take by using a computer to study behavior, is the key to understanding the material.

Now you know what a dynamical system is, but what are we trying to do with them? We said before that we are interested in understanding the long term behavior of these systems. Mathematically, this means we want to answer this question: if we are given a number, and we apply our function to it over and over, what happens? Of course, this may depend on which number we are given. For the square root function you used above, you may have noticed that eventually number displayed by your calculator stops changing. Write down the number you see. Repeat the whole process with a different starting number. Keep trying this with different numbers until you have a conjecture about the long term behavior under the square root function.

What happens if you use some function other than the square root? What sort of behavior do you get by repeatedly taking the cosine of a number? The square of a number? What about repeatedly pushing the "1/x" or "+/-" buttons?

The repeated application of a function is called iteration. Here is a formal definition.

**Definition (Iteration):** For a function f and a point y, f(y) is
called the *first iterate* of y. f(f(y)) is called the *second
iterate* of y. Repeatedly evaluating the function like this is
called *iteration*.
**Definition (Orbit):** The set of all iterates is called the
*orbit* of y.

F(P)= (1+r) P

What about after two years? Since the bank compounds annually, we need
to include in the second year, the money earned in interest in the
first year. Thus the interest earned after two years is the *second
iterate* of the function F. The amount of money earned after n
years is the nth iterate of F.

G(P)= (1+r) P (1 - b P).

It would be nice to know how population changes over many years. Does it die out, does it explode, or is it something in between? In other words, we are interested in the long term behavior of points under iteration by G. In general, this depends on the variables r and b. In the following section, we study iteration under a similar set of equations. Keep the idea of population in mind and think about the implications of your findings related to such a model.

Iterated functions come up in a variety of mathematical models. In order to better understand these models, we will look at how the long term behavior changes.

**Definition (Fixed Point):** For a function g(x), a point p is a
*fixed point* when g(p)=p.

- What are the fixed points of f(x)=x^2?
- Pick a point "near" (distance less than .1 is near enough) a fixed point. Keep iterating (pushing the squaring button) until you see a pattern for the long term behavior. Try again with other numbers near each of the fixed points. Write down what happens for each choice. Does it matter if the number you pick is less than or greater than the fixed point?

Notice that for the squaring function, some orbits converge to a point. Some orbits diverge (get arbitrarily large). Are there orbits which always stay bounded, but do not converge to just one point? In general, the answer is yes. You will observe more complicated long term behavior of orbits in your computer investigations.

**Definition (Attracting and Repelling Fixed Point):** A fixed
point p is easy to detect computationally when orbits of nearby points
converge to p. In this case, p is called *attracting*. If the
orbits of nearby points move away from p, p is called
*repelling*. A fixed point which is neither attracting nor
repelling is called *neutral*.

- Which of the fixed points for the squaring function are
attracting? Which are repelling?
- Every point of the identity function i(x)=x is a neutral fixed point. Do you see why?

Author: Evelyn Sander

Comments to:
webmaster@geom.umn.edu

Created: Jun 09 1996 ---
Last modified: Jul 31 1996