- Periodic points

- Eventual behavior

- Hartman and Grobman

- Algebraically verify that +1 and -1 are fixed points of the
function
*f*(*x*) = 1/*x*.

- Algebraically verify that 0 is a fixed point of the "+/-"
function.

- Algebraically find the fixed points of the square root function.

Let *f*^{n} denote the *n*-th iterate
of *f*. For example, *f*^{2}(*x*) =
*f*(*f*(*x*)) is the second iterate of
*x*.

**Definition(Periodic Point):**
We say that *p* is *periodic with least period*
*n*, if *f*^{k}(*p*) =
*p* for *k = n*, and this is not true for any smaller
value of *k*.

The orbit of a periodic point *p* with least period *n*
consists of exactly *n* different points, since
*f*^{k}(*p*) = *p*. Saying
"least period" all the time is cumbersome. Therefore, whenever we
refer to a period *n* point, it is assumed to be of least
period *n*.

For example, 1 is a fixed point of the function *f*(*x*)
= 1/*x*, but 3 is mapped to 1/3 and 1/3 is mapped back to 3, so
3 is a periodic point of *f*(*x*) = 1/*x*, and
its least period is 2. Note that 1/3 is then a periodic point of least
period 2 as well.

- What is the period of a point under application of the "+/-"
button?
- Check that for
*f*(*x*) = 3.2*x*(1 -*x*), the points*x*= 0 and*x*= 11/16 are fixed points. Check that and are period 2 points.

*Points of period 2 for the Logistic map.*

The method of finding fixed points graphically also works to find
periodic points. For example, to find the points of least period two
for a function *f*, look for the intersection points of the
graph of *f*^{2} with the line *y = x*. The only
catch is that fixed points are also period two points, but not of
least period two. Thus you must first find the fixed points for
*f* and discard them when looking at intersections of the graph
of *f*^{2} and the line *y = x*.

For the following questions, read the instructions for the Macintosh or for other machines.

- Draw the graphs of the first and second iterate of the Logistic
map for = 3.2.
Check graphically the number of points of least period 2. Does this
agree with what you found above analytically?

- Graphically determine the number of period two points for functions in the Logistic family at = 2, 3.2, and 3.5. Remember to exclude fixed points when you count intersections.

Just as with fixed points, periodic orbits can be attracting, repelling, or neutral. For a given periodic orbit, if orbits of nearby points converge to the periodic orbit, it is attracting. If orbits of nearby points move away from the periodic orbit, it is repelling. Otherwise it is neutral.

- Periodic points

- Eventual behavior

- Hartman and Grobman

Written by Hinke Osinga

Comments to:
webmaster@geom.umn.edu

Created: Apr 3 1998 ---
Last modified: Thu Apr 9 13:49:41 1998