Due: Before Spring Break. More precisely, your report must be submitted by 10 a.m. on Friday, March 1.

Group project: Your group is limited to at most four members. Once a group begins work on the project, its membership cannot change. Consequently establishing your group must be your first step in this project. Each group will submit one report, and all members of the group will receive the same grade for this project.

In the past I have observed that groups with only one or two members do not do as well as groups with four members. Consequently, you will need my permission to work in a group with fewer than three members. Before I grant such a request, I will try to help you make contact with other students in the class who might be compatible group members. If you are not in a group with at least three members, you must send me email by February 20.

Goals:

1. To become familiar with the long-term behavior of solutions to certain second-order equations.
2. To get experience doing numerical work with differential equations.

Second-order equations: Autonomous, second-order differential equations are often studied numerically by reducing them to first-order systems with two dependent variables. In this project, you will use the computer to analyze a nonlinear second-order equation with parameters. You will analyze phase portraits and theta(t)- and v(t)-graphs to describe the long-term behavior of the solutions for certain parameter values, and you will collect numerical evidence that suggests certain bifurcation values.

As discussed in class, the equation of motion of a (damped) pendulum is theta" + b theta' + k sin(theta) = 0, where theta is the angle that the pendulum rod makes with the vertical, b is a damping coefficient, and k is a constant that is determined by the length of the rod. In your report, you should describe the motion of the pendulum assuming certain values of b and k. (See below regarding the value of k for your group.)

Your report should discuss the following items:

1. (no damping) First you should study the equation in the absense of damping. In other words, b=0. Examine solutions using both their graphs and the phase portrait. Describe the behavior of solutions that have especially different long-term behavior, and be specific about the physical interpretation of the solutions.

   Approximate the periods of periodic solutions as a function of the initial conditions and discuss your results.

2. (linear approximation) It is common to approximate the sine term in the equation with its linearization at theta=0. In this case, the equation becomes
   theta" + b theta' + k theta = 0.

   Repeat the analysis requested in Part 1 and compare the results you obtained there with these results. How are the phase portraits similar and how do they differ?

3. In Parts 1 and 2 above, you considered the undamped case. Now you are going to experiment with the damping parameter b and study how it affects the behavior of the solutions. First, compare your results in Parts 1 and 2 with the behavior of solutions if b = 0.1. What aspects of the solutions change? What aspects are similar to those of the undamped case? Next, increase the damping dramatically by considering b = 5.0. What behavior remains the same? What changes?

   Now consider how varying the parameter b between 0.1 and 5.0 changes the behavior of the solutions. In fact, there is an important parameter value that separates the ``b = 0.1-like'' behavior from the ``b = 5.0-like'' behavior. This parameter value is difficult to locate numerically using only the computational aids that are currently available to us. Nevertheless, do your best to locate this bifurcation value b = b₀. Describe the behavior of the solutions for parameter values slightly less than b₀ and for parameter values slightly larger than b₀.

Your report: Your report should be no longer than four typewritten pages, and it should address all of the questions mentioned above. You may provide as many illustrations from the computer as you wish, but the relevance of each illustration to your report must be evident. Illustrations are not included in the four-page limit. (Please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are worth nothing.) Please insert your illustrations at appropriate places in your report rather than attaching them to the end of the report.

Parameter values: The value of k that you should use is determined by the last digits of the BU ID numbers of the members of your group. Use k = 0.1 a + 1.05 where a is the average (accurate to two decimal places) of the last digits of all members in the group. For example, if the last digits are 0, 1, 2, and 2, then a = 1.25 and k = 1.175.

Academic Conduct: Your work and conduct in this course are governed by the CAS Academic Conduct Code. This code is designed to promote high standards of academic honesty and integrity as well as fairness. Copies of the code are available in CAS Room 105, and it is your responsibility to know and follow the provisions of that code. In particular, all work that you submit in this course must be your original work. For example, the computations that you do for your report as well as the text of your report must be original to your group. All group members are responsible for all aspects of the report. Any cases of suspected academic misconduct will be referred to the CAS Student Academic Conduct Committee.