Calc III Lab #8: Questions
The system of ODEs that we will study is given by
dx/dt = v
dv/dt = -x + x^2 -damping v + tau.
Here damping>0 is a parameter called the coefficient of damping.
It reflects
the amount of dissipation in the system. If damping=0,
then there is no friction.
The parameter tau affects the size of the region in which
oscillations can occur. The notation x^2 means "x
squared."
Your goal is to find out as much as possible about this system!
In your lab report, you should clearly
explain the dynamics of this system as completely
as you can. The more you explain, the more points you will get.
You may make up your own questions and may report
on any properties of the differential equations, but
to receive full credit, you should use
numerical (DsTool), symbolic (Maple) and analytical
techniques (from lecture) to answer at least the following questions:
Question #1
Produce phase portraits for the parameter values below and
indicate the phase portraits on the graph provided.
(One of the phase portraits is already indicated as an example.)
- (damping,tau)=(0,0)
- (damping,tau)=(0,0.5)
- (damping,tau)=(0.5,0)
- (damping,tau)=(0.5,0.5)
Now go on to the next question. If you have time, come back to this question
and try a few other phase portraits. For example:
- (damping,tau)=(0.5,0.1)
- (damping,tau)=(0,0.1)
- (damping,tau)=(0.5,-0.5)
Question #2
Compare the phase portraits with damping=0
to the graph of the function
Hint: Consider the level sets (or contours) of the surface for
tau=0, 0.1, and 0.5. This is most easily done in
Maple using the plot3d command with the
style=PATCHCONTOUR option. You may want to rotate the
Maple plot to make it look like the DsTool phase portrait.
- What surface features do the equilibria correspond to?
- What surface features do trajectories correspond to?
- Extra Credit: Explain what happens to the surface as tau
increases past tau=0.25.
(Hint: You may wish to answer question 3.4 first.)
- Extra Credit: Can you determine where this surface "came from"?
In other words, can you somehow derive the fact that
the geometry of this surface is related to trajectories of the ODE?
(Hint: We saw something like this a few class periods ago;
use the chain rule on the quantity dv/dt.)
Question #3
Complete these questions at home; Please show all your work.
Determine the location and stability of equilibria. In other words:
- Determine the location of equilibria in terms
of the parameters. For what values of the
parameters are there exactly two equilibria?
- Linearize the ODE about each equilibrium.
To do this:
- Compute the Jacobian of the right-hand side of the ODE.
- Evaluate the Jacobian at the location of the first equilibrium.
- Use this matrix to write down the linear system that
approximates the nonlinear system in a neighborhood
of the first equilibrium.
- Repeat the previous two steps for the second equilibrium.
- Based on the linear analysis and the "pitchfork diagram," determine the
stability of each equilibrium. Do your results agree with the
phase portraits you generated?
- For what values of the parameters do bifurcations occur? Indicate the
bifurcation curves on the graph provided.
Because of the Winter Break, please mail in you lab reports
(postmarked by Tuesday, December 13, 1994) to:
Jeremy Case
127 Vincent Hall
Department of Mathematics
University of Minnesota
Minneapolis, MN 55455
Go To
Frederick J. Wicklin <fjw@geom.umn.edu>
Last modified: Mon Dec 5 17:02:32 1994