MA 226: LAB 5

The Glider Equations

This lab is due Tuesday, May 2, 2000, by 7 PM. Turn it in in class or to the TF in the Mac Lab. I suggest that you use the IDE software in the Mac Lab. You will be graded on exactly what is asked for in the instructions below. You need not turn in any additional data, graphs, paragraphs, etc. You should submit only what is called for, and in the order the questions are asked. It is perfectly acceptable to hand in hand-drawn figures, since you cannot print pictures using IDE. Remember that you will be graded on your use of English, including spelling, punctuation, logic, as well as the mathematics.

IMPORTANT: The work you submit should be your own and nobody else's. Any exceptions to this will be dealt with harshly. Copying or paraphrasing another student's lab report is strictly forbidden.

Lab. In this lab you will investigate a nonlinear system of differential equations that governs the motion of a toy glider (you may remember the light, balsa wood gliders that made loop-de-loops when you threw them). This is a two dimensional system of equations given by

dx/dt = (-cos(x) + v2)/v
dv/dt = -sin(x) -Dv2

Here D is a parameter that you will vary between 0 and 4. D is called the drag for the system. We will choose the variable v to be positive. As you can see from the equations, the variable x is periodic with period 2 pi.

Note: In the IDE lab called the glider, the variable x is called theta.

1. By looking at both the phase plane and the animation of the glider's motion, tell explicitly what the variable x corresponds to in the physical model.

2. Find all equilibrium points for this system when D = 0. Use the method of linearization to compute the eigenvalues of the linearized system at the equilibrium points.

3. What is the motion of the physical system corresponding to the equilibrium point(s)?

4. Now let D > 0 . Find all equilibrium points and their linearization. Is there any bifurcation that occurs when D becomes slightly positive? Hint: In these computations you will encounter terms like sin(arctan(#)). Remember how to simplify these expressions: draw a triangle with an angle that has tangent = #, then find the sine of that angle.

5. What happens as the parameter D passes through the value marked Dc? What is the exact value of Dc?

6. When D = 0, there is a certain regime in the phase plane where the glider does "loop-de-loops." There is another region where the glider does not loop. Sketch carefully these two regions, showing the solution curves in each. Also sketch the boundary between these two regions. What happens to the plane when its solution curve lies in this boundary?

7. What happens to the glider when D > 0?