MA 231: Multiple Bifurcations Lab

A Two Parameter Family of Differential Equations

This lab is due **Thursday, October 2, in class. **
You may use any technology you wish
or no technology at all to perform this lab. Perhaps easiest,
however, is to use the tool called Phase Lines that comes with the book.

In your report, you should submit **only** what is called for below, and
in the order the
questions are asked. It is perfectly
acceptable to hand in hand-drawn figures,
in case the queues at the printers become too long.
You need not turn in any additional data, graphs, paragraphs,
etc.
Remember that you will be graded on your use
of English, including spelling, punctuation, logic, as well as the
mathematics.

**IMPORTANT:** The work you submit should be your own and nobody else's.
Any exceptions to this will be dealt with harshly.

In this lab you will investigate the bifurcations that occur in the two parameter family of differential equations given by

This differential equation depends upon two parameters, **r** and
**a**. Your goal is to investigate the bifurcations that occur as
you hold one of the parameters in this equation fixed and vary the
other parameter.

Answer each of the following questions about this family of differential equations.

**1.** Fix ** r = 1** and let **a** vary. In one or two
paragraphs, complete with pictures,
describe the bifurcation that you see. Include a picture of
the bifurcation diagram. Explain how you know that this bifurcation
takes place. (HINT: you cannot solve these equations easily, so use
whatever means to draw the graphs of the right hand side of the
equation to determine the equilibrium solutions.

**2.** Now fix ** r = -1** and let **a** vary, and then repeat
question 1.

**3.** Now fix ** r = 0** and let **a** vary, and then repeat
question 1. (This time you can solve by hand).

**4.** Does the bifurcation diagram in question 1 change
significantly if you
choose any positive value of **r**? Why or why not? Does the
bifurcation diagram in question 2 change significantly if you
choose any negative value of **r**? Why or why not?

**5.**Now fix the value **a = 2** and vary the parameter
**r**. Describe any bifurcations that you see. Draw the
bifurcation diagram for this family.

**6.** Repeat question 5, this time for **a = -2**.

**7.** Does the bifurcation diagram in question 5 change
significantly if you
fix any positive value of **a**? Why or why not? Does the
bifurcation diagram in question 6 change significantly if you
choose any negative value of **a**? Why or why not?

**8.** The bifurcation diagram you find in question 5 should be
markedly different from that in question 6. For which value of
**a** does the bifurcation diagram change? Explain why in a
paragraph with a few pictures and graphs.

**9.** Finally, draw a picture of the **r, a** plane. In this
picture draw different regions where the phase line corresponding to
the given values of **r** and **a** are qualitatively the same.
Explain what your diagram means carefully.

Have fun!