Due: Before Spring Break. More precisely, your report must be submitted by 10 a.m. on Friday, March 3.

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.

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 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 planes and x(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.

This contraption consists of a flexible steel beam attached to the top center of a box. There are magnets on either side of the bottom of the box toward which the beam is attracted. The differential equation describes the motion of the tip of this beam. The differential equation for the cantilever beam is x" + bx' + kx + x3 = 0. Here b is the damping coefficient, and k is related to the stiffness of the beam and the strength of the magnets. The center of the box is located at x = 0, with x > 0 to the right. In your report, you should describe the motion of the beam assuming certain values of b and k. (See below regarding the value of k for your group.)

Your report should discuss the behavior of the solutions in the following cases:

1. (weak magnets with no damping) First you should study the equation in the absence of damping. In other words, b=0. Examine solutions using both their graphs and the phase plane. Are the solutions periodic? If so, approximate the period of the solutions. Describe the behavior of a few solutions that have especially different initial conditions. Be specific about the physical interpretation of the different initial conditions.

   Consider the quantity H(x,v) = (x⁴)/4 + k(x²)/2 + (v²)/2. Using the multivariate Chain Rule, show that the value of H remains constant throughout any particular motion of the beam. What does the function H tell you about the solution curves of the system?

2. (weak magnets with damping) In Part 1 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 Part 1 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.

3. (strong magnets with no damping) Once again let b=0, but now use -k in place of the parameter k. Repeat your analysis in Part 1 giving a complete description of the phase portrait.

4. (strong magnets with damping) Now bring back the damping term and determine the bifurcation value b₀ for this equation. Give a complete description of the phase portrait both for values of b such that 0<b<b₀ and for values of b such that b>b₀. What parts of the phase portrait separate the different possible behaviors of the beam?

Your report: Your report should be no longer than five 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 five-page limit. (Please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are worth nothing.) Examples of good reports done here at BU in previous semesters are available for inspection in my office.

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 + 1.07 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 = 1.195.

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.