Due: Your report must be submitted by 9:30 a.m. on Thursday, November 2.

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 two different second-order equations with parameters. You will study their 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. In Sections 2.1 and 2.3 we discuss the (damped) harmonic oscillator

as a model of a mass-spring system. Here m is the mass, b is the damping coefficient, and k is the spring constant. Throughout this project we are going to assume that m=1. In your report you should describe the motion of the system assuming certain values of b and k. You will also describe the motion of a somewhat related system that includes this mass-spring system as well as a "one-sided" spring. You can imagine the one-sided spring as a rubber band.

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

1. (Ideal mass-spring system with no rubber band) Using the spring constant k₁ as determined below, you should study the equation in the absense of damping. In other words, b=0, and the equation is
   x" + k₁x = 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.

2. (Mass-spring system with damping but no rubber band) 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 = 1.0. What aspects of the solutions change? What aspects are similar to those of the undamped case? Next, increase the damping dramatically by considering b = 10.0. What behavior remains the same? What changes?

   Consider how varying the parameter b between 1.0 and 10.0 changes the behavior of the solutions. In fact, there is an important parameter value that separates the ``b = 1.0-like'' behavior from the ``b = 10.0-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. (Mass-spring system with a rubber band but no damping) Once again let b=0, but now add a rubber band to the system. As discussed in class, this results in a more complicated equation of the form
   y" + k₁y + k₂y⁺ = 10

   where y⁺ represents the function that is equal to y if y>0 and 0 otherwise. In MATLAB you can represent the function y⁺ using the boolean notation:

   y*(y>=0)

   Repeat your analysis in Part 1 giving a complete description of the phase portrait for this system.

4. (Mass-spring system with a rubber band and damping) Now add damping so that the equation becomes
   y" + by' + k₁y + k₂y⁺ = 10.

   Repeat your analysis in Part 2 for this equation.

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.

Examples of good reports done here at BU in previous semesters are available for inspection in my office.

Parameter values: The values of k₁ and k₂ that you should use are determined by the last digits of the BU ID numbers of the members of your group. Use k₁ = 12 + 0.1 a and k₂ = 5 - 0.05 a 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, k₁ = 12.12, and k₂ = 4.94.

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.