- What are the fixed points of
*f*(*x*) =*x*^{2}?

What is the relation between the fixed points of the square function and the intersection points of the graph of the square function with the special lines*y = x*and*y = -x*?

- Consider the square function on your calculator. Pick a point
"near" (distance less than 0.1 is near enough) a fixed point. Keep
iterating (pushing the squaring button) until you see a pattern for
the long term behaviour. 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?

- Which of the fixed points for the square function are attracting?
Which are repelling?

## Questions on Part 3

#### Iterates versus time

- Use the Java Applet
Iterates versus time for
the Logistic map to see how the plot changes as
varies. Describe
what you see.

Note that always the iterates of*x*_{0}= 0.5 are taken. Can you find values for*x*_{0}( 0) for which the orbit behaves different from the orbit of*x*_{0}= 0.5?

#### Graphical iteration

- Use the Java Applet Iteration
of the Logistic Map to see how the dynamics close to the
(positive) fixed point depends on the parameter
. When you start
the applet, this fixed point is attracting. Is it attracting for all
*x*_{0}? Is it attracting for all ? If not, what happens right after it stopped being attracting?

- Use the Maple package
for this question.

Fix the starting point*x*_{0}= 0.1. For several choices of between 0 and 3.5, investigate the eventual behaviour of*x*_{0}= 0.1. In the Java Applet you can click on the*clear*button to see what the eventual behavior is; in Maple the eventual behaviour is shown in red.

Draw a diagram with on the horizontal axis and*x*on the vertical axis. For each value that you checked, place a dot at the*x*values that are part of the eventual behaviour of*x*_{0}= 0.1. If you checked the eventual behaviour for enough values, you should be able to sketch the curves connecting the dots, showing the*continuous*dependence on . Do not go beyond = 3.55.

## Questions on Part 4

#### Periodic points

- 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.

- 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*x*= and*x*= are period-2 points.

- 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 in the previous question?

- Graphically determine the number of period-2 points for
functions in the Logistic family at
= 2, 3.1, and
3.5. Remember to exclude fixed points when you count
intersections. Are they attracting?

#### Eventual behaviour

- Using the techniques from the part on
Eventual behaviour, draw a
qualitative diagram for the eventual behaviour of all points under the
Logistic map for
= 2. Your diagram does not need to have exact values of fixed and
periodic points. The idea is to compare pictures qualitatively.

- Using the techniques from the part on
Eventual behaviour, draw a
qualitative diagram for
= 3.1. As you
investigate, magnify the region near the fixed point other than 0.
Compared to the behavior for
= 2, what has
happened?

- Using the techniques from the part on
Eventual behaviour, draw a
qualitative diagram for
= 3.5. What is
the period of the attracting orbit here? Indicate in which order the
points in the orbit occur.

#### Hartman and Grobman

- Compare the figure for
= 2.5 with the
figure for
= 1.5. What is
the difference in behaviour of the graphical iterations? Why do you
think this difference is there? (Hint: look at the slope of the graph
at the attracting fixed point.)

- Recall that in order to find period-2 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-2 points are attracting, repelling, or neutral?

- Describe geometrically how your relationship (between the slope of
the graph of the second iterate of the function and whether the
period-2 points are attracting, repelling, or neutral) might
generalize to period-
*n*points.

## Questions on Part 5

#### Qualitative change of dynamics

- Recall that
*p*is a fixed point of*f*if*f*(*p*) =*p*. Verify that the Logistic family*f*(*x*) =*x*(1 -*x*) has fixed points*x*= 0 and*x*= .

The slope of the graph of the Logistic map at a fixed point*p*can be computed as follows: Determine the derivative of*f*with respect to*x*. Then replace any*x*in the derivative with*p*. This expression is exactly the requested slope.

Compute the slope of the graph of the Logistic map at the fixed point 0 in terms of . What is the slope for the other fixed point?

- For what values of
are the fixed
points of the Logistic family attracting? Repelling? Neutral? Test
your values using the computer.

#### Bifurcation points

- Approximately at what
value between
1.5 and 3.1 does a bifurcation occur?

- At exactly what
value between
1.5 and 3.1 does a bifurcation occur? Explain why you know this value
is exact.

#### Orbit diagram

- Consider the orbit diagram for the Logistic map. At
= 1 the curve
suddenly turns away from the horizontal axis. What has happened?

- Somewhere between 2 and 3.1, the orbit diagram curve splits into
two branches. 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
increases.
Describe the sequence of bifurcations. In other words, how does the
period of the attracting set change at each bifurcation?

## General Questions on One-Dimensional Dynamical Systems

- In Maple, type
`read(`Quadratic.mpl`):`

after you loaded the package`Chaos.mpl`

. Your dynamical system is now

*f*(*x*) =*x*^{2}+ .

Investigate the behaviour of the iterates for parameters in a small interval around = -0.75. Discuss the bifurcation that occurs in the Quadratic family for = -0.75.

- In Maple, load the
package
`Chaos.m`

, then type`read(`Sine.mpl`):`

. Your dynamical system is now

*f*(*x*) =*Sin*(*x*).

Investigate the behaviour of the iterates for parameters in a small interval around = 1. Discuss the bifurcation that occurs in the Sine family for = 1.

- Using Maple, compare
the orbit diagrams for the Logistic family
*f*(*x*) =*x*(1 -*x*) and for the Quadratic family*f*(*x*) =*x*^{2}+ . Do you see any similarities? Discuss them and point out the differences.

What is the range of values in the diagram for the Logistic family? What is the range of values in the diagram for the Quadratic family?

For = 4, the graph of the Logistic map is special. The interval [0,1/2] is mapped entirely onto [0,1] and also the interval [1/2,1] is mapped entirely onto [0,1]. The Logistic map covers the interval [0,1] twice. Notice that this is not true for smaller (positive) values of .

Find the value for such that the graph of*f*(*x*) =*x*^{2}+ covers the interval [-2,2] exactly twice, as described above for the Logistic map.

Copyright © 2001 by Hinke Osinga

Last modified: Tue Nov 11 10:49:30 2003