*This picture shows graphical iteration for a member of the quadratic family.*

- Double click on the x value for which you want to start the
iteration. Your choice will appear in the
*Analysis Start*box. - Alternatively, edit the
*Analysis Start*box to your desired initial x value.

- Since we are interested in long term behavior, we only care
about how the iterates behave after a long time. The initial iterates
are irrelevant for long term behavior, and are called
transients. Click on the
*Transients*button to remove these transients. Clicking on it again toggles back to seeing all iterates. - To get a closer look at a region, use the mouse to select a
region, and click on the
*Magnify*button.

From Part 1, which of the fixed points are attracting for c=0? Repelling? Here is a picture representing the eventual behavior of all points on the line for c=0. The arrows represent the direction that points go after iteration.

- Draw a qualitative diagram for the eventual behavior of all
points for c=-0.4. Your diagram should resemble the picture above, but
it does not need to have exact values of fixed and periodic
points. The idea is to compare pictures qualitatively. Use the
techniques from the Methods to Study Iterates
section above to formulate the diagram.
- Draw a similar diagram for c=-1.1. As you
investigate, magnify the region near the smaller fixed point. What has
happened? What is the period of the attracting set? In some way,
indicate the period of all the periodic points. Also remember to indicate what
happens to large negative values of x.
- Draw a similar diagram for c=-1.3. What is the period
of the attracting orbit here? Indicate in which order the points in
the orbit occur.
- What happens for c=-2?

Now that you have investigated dynamics for some select values of c, think about the following ideas for the entire family f_c.

The dynamics of a function is a qualitative description of the eventual behavior of all points on the line. This includes information such as the number and relative placement of fixed and periodic points and whether they are attracting, repelling, or neutral. You should have found that at each of the values c= -0.4, -1.1, -1.3, and -2, the dynamics is qualitatively distinct.

*Definition (Bifurcation):* A change in the dynamics for a
certain c is called a *bifurcation*.

- Does every value c give different dynamics? Look at some intermediate
values between c=-0.4 and c=-1.1. Is the dynamics at each intermediate
value qualitatively distinct from the dynamics you have already
studied? If not, what happens?
- Between c=-0.4 and c=-1.1, at approximately what c value does a bifurcation occur?

Author: Evelyn Sander

Comments to:
webmaster@geom.umn.edu

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