Due: 9:30 a.m. on Tuesday, March 31.

Group project: This is a group project with each group having either three or 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.

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.

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 bifurcation value b₀ that separates the ``b = 0.1-like'' behavior from the ``b = 5.0-like'' behavior. This bifurcation value is difficult to locate numerically using only the computational aids that are currently available to us, but try your best. Compare the behavior of the solutions for parameter values slightly less than b₀ to those 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 part of the four-page limit. Please insert your illustrations at appropriate places in your report rather than attaching them to the end of the report. Examples of good reports done here at BU in previous semesters are available for inspection during office hours and discussion section.

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

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.

Asking questions: Questions by email about the project are not allowed. Instead we will use the Blackboard Discussion Board for this course. I will check that board for questions once a day. The last time that I will check the discussion board will be noon on March 29. Do not wait until the last minute to ask questions. Please start a new thread with each new question.

Academic Conduct: Your work and conduct in this course are governed by the Boston University Academic Conduct Code. This code is designed to promote high standards of academic honesty and integrity as well as fairness. It is your responsibility to know and follow the provisions of the code. In particular, all work that you submit in this course must be your original work. For example, you can only discuss your project with other members of your group, with Professor Blanchard, Brian, or Rob. Moreover, 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. If you have a question about any aspect of academic conduct, please ask.