Due: Your report must be submitted by 10:00 a.m. on Friday, April 18.

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 forced second-order equations.
2. To study the relationship between the frequency of the forcing and the amplitude of the solutions for the harmonic oscillator.
3. To study the relationship between the amplitude of forcing and periodicity of solutions in a nonlinear equation.

Parameter values: There are two parameter values k₀ and b₀ that are determined by the last (single) digits of your BU ID numbers. The parameter k₀ is used in Parts 1 and 2, and the parameter b₀ is used in Part 3. The values of these two parameters are the same, but I've used different notation to emphasize the fact that their meanings are dramatically different. Use k₀ = 5 + 0.1 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 and k₀ = 5.12. Likewise, b₀ = 5.12.

Second-order equations: Using all of the methods that we have developed in this course, you will analyze the long-term behavior of the solutions to two forced, second-order equations.

We begin with the forced harmonic oscillator

Here b is the damping coefficient, k₀ is the spring constant (based on your ID numbers), and w determines the frequency of the forcing.

1. (An undamped, forced linear equation) First consider the equation in the case where b=0. Using the Method of the Lucky Guess, determine a particular solution to the equation. Then using a numerical solver and/or the formulae for the solutions, estimate the amplitudes of the solutions for frequencies w in the interval 0 <= w <= 3. How do the amplitudes in the nonresonant case relate to the amplitudes in the resonant case?

2. (Damped, forced linear equations) Assume that the forcing is present along with some damping (b>0). How long does it take any given solution to get close to the steady-state solution? Using a numerical solver and/or the formulae for the solutions, estimate the maximum amplitudes of the solutions for frequencies w in the interval 0 <= w <= 3 for various values of b. Graph the maximum amplitude as a function of frequency for different values of b, all on the same set of axes. What happens to these graphs as b approaches 0?

Now we return to a variation on the nonlinear equation (equation 4) that we studied in the first project. The equation is

3. Search for periodic solutions to this forced equation for values of e in the interval 0.01 < e < 0.66. For each periodic solution, specify the value of the parameter e, the initial condition, and the period. (See the comment below regarding the fact that periodic solutions come in groups.)

Some history: The equation in Part 3 is called the forced van der Pol equation. Van der Pol studied this equation as a model for a vacuum tube triode circuit approximately 80 years ago. He and his colleague, van der Mark, noted the existence of two stable periodic solutions with different periods for certain values of the parameter e. Their work inspired two English mathematicians, Mary Cartwright and J. E. Littlewood, to study this equation in the 1940's and 50's, and they were able to prove that there are intervals of parameters for which there are two stable periodic solutions with different periods. These parameter values yield "chaotic" solutions. Since this work was roughly fifteen years prior to Lorenz's famous work on the Lorenz attractor (see Section 2.5 of our book), one could claim that chaos theory actually started with the work of Cartwright and Littlewood.

Even though the existence of these stable periodic solutions has been known for almost 60 years, there are relatively few graphs of these solutions in the literature (as far as I can tell). Here's one from a paper of Littlewood published in 1963. Here's one of period 18 pi that I found last Saturday:

Finding periodic solutions of a forced equation is more difficult than finding periodic solutions of autonomous equations. In the autonomous case, any solution curve that comes back to where it started in the phase plane corresponds to a periodic solution. Unfortunately, in the nonautonomous case, the vector field varies in time, and solution curves in the phase plane can cross.

It is a little tricky to use a computer to tell if you have found a periodic solution, but for this equation you can take advantage of the fact that the equation is periodic in t with period 2 pi. So if a solution goes for 2 pi units of time and ends up back at the same initial condition, then the solution has to be periodic. (Why? This point of view is discussed on pp.518-529 of our text.) The same observation holds for any integer multiple of 2 pi. (Note that my solution has period 18 pi.) Also, if you find a periodic solution with a period of 2 k pi for k>1, then you have actually found k different periodic solutions because you can translate any such periodic solution by any multiple of 2 pi and get another solution. (Why?) So my periodic solution above actually represents 9 different periodic solutions.

HPGSystemSolver can help you with your search, but don't use the default step size. For this differential equation, I recommend that you set the step size to 0.001.

Attempting this part of the project with HPGSystemSolver alone is somewhat like building a house with only a hammer and a hand saw. It's possible, and people have done it. But there are easier ways. If you are familiar with one of the computer math packages such as Maple, Mathematica, or MATLAB, you can save yourself a lot of tedious guesswork.

Your report: Your report should be no longer than four typewritten pages, and it should address all of the questions in the project description. You may provide as many illustrations from the computer as you wish, but the relevance of each illustration to your report must be evident. (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.

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. A copy of the code is available in CAS Room 105 if you cannot access it on the web, and 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, 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.