- Verify that the quadratic family f(x)=x^2+c has fixed points of
the form

.5 (1+ sqrt(1-4c)) and .5(1-sqrt(1-4c)). - For a variety of values of c, compare the graphs of functions in
the quadratic family to the identity map. Use this to determine where
these functions have fixed points. For each c, how many fixed points
are there? Are there the same number of fixed points for every c?
Relate your findings to the previous problem.
- Use Chaos Lab #2 to graph the second iterate of the quadratic
family. To do this, change the value in the
*Iterate*column from 1 to 2. For parameter c=-1, check graphically the number of points of period two. You may wish to simultaneously graph the first and second iterates to see which points are least period two.#### Questions from Part 3.

- Draw a qualitative diagram for the eventual behavior of all
points under the quadratic map for c=-0.4. Your diagram does not need
to have exact values of fixed and periodic points. The idea is to
compare pictures qualitatively.
- 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?
- 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?
#### Questions from Part 4.

- For a map in the quadratic family, what is the slope of its
graph at the fixed point?
- Iteration of a linear map with this slope is similar to
iteration of the quadratic function near the fixed point. For what
values of c should fixed points for the quadratic family be
attracting? Repelling? Neutral? Test your values using the Chaos Lab
#2.
- State the relationship between slope of the graph at the
fixed point and whether the fixed point is attracting, repelling, or
neutral.
- Recall that in order to find period two points, you graphed the
second iterate of the map. Can you find a relationship between the
slope of the graph of the second iterate of the function and whether the
period two points are attracting, repelling, or neutral?
- Describe geometrically how your relationship might generalize
to period n points.
#### Questions from Part 5.

- In Chaos Lab #3, between -0.4 and -1.1, the orbit diagram curve
splits into two branches. How does this relate to what you saw in
Chaos Lab #2? Why are there two branches now? What do they correspond
to? (Hint: What was the eventual behavior of points in this region?)
- Investigate further bifurcations as c decreases. Describe the
sequence of bifurcations. In other words, how does the period of the
attracting set change at each bifurcation?
- How would you describe the orbit diagram near c=-2?
#### General dynamical systems questions

- The line x=0 divides the plane in half. Label the
half "+" and "-", corresponding to positive and negative x values. For
c=-2, pick a point, iterate four times. Write down a series of +'s and
-'s corresponding to whether the x-value for the point and the
iterates are positive or negative. This ordered list of +'s and -'s
is called the
*itinerary*for the point. - Find a point which starts in the + region, and with first iterate
in the + region, second iterate in the - region. How close do you have
to be to this point to stay in these same regions under iteration? Try
the same thing for a point with itinerary +,+,-,+. Again, how close do
you have to be to this point to have the same itinerary? Try again
for the itinerary +,+,-,+,-.
- Based on the previous two questions on itinerary, for a physical
system modelled by the quadratic family, make a statement about your
ability to predict the future.
- The pattern of bifurcations you have concentrated on here is called the period doubling sequence. Can you see any other patterns of bifurcations? To help think about this, look at the strange way of ordering the integers described in the Geometry Forum article Sharkovskii's Theorem.

Author: Evelyn Sander

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Created: Jun 26 1996 ---
Last modified: Jul 31 1996