Calc III Lab: Foci
The phase portraits you will see in this section of the lab are called
foci. The equilibrium in the phase portrait for a focus is
sometimes referred to as a "sink" (when nearby orbits are attracted to
it) or a "source" (when nearby orbits are repelled by it).
Activity
Compute solutions to the linear differential equation determined by
the matrix with entries
(a,b,c,d)=(-1,-1,0.5,0).
- Clear the old phase portrait.
- Compute the phase portrait for this system.
- Compute the trace and determinant of the associated matrix.
- Plot the ordered pair (det, trace) on the diagram provided.
- Sketch the phase portrait on the diagram. Again,
you may want to put arrows on the phase portrait
to indicate the direction that trajectories flow.
- Indicate any straight-line solutions, or write "No straight-line
solutions" on your sketch.
- Clear the old phase portrait.
- Do the same activity for the linear differential equation determined by
the matrix with entries
(a,b,c,d)=(1,-1,0.5,0).
Question 1
- How are the phase portraits for foci similar to the phase portrait
for a center?
- How are they different?
- Describe a physical or biological situation in which can be modeled by
a differential equation whose phase portrait could be either
a center or a foci. Specify the quantities that are
evolving in time, and indicate how the "swirling" phase portrait
indicates oscillations in your quantities.
Go To
Robert E. Thurman <thurman@geom.umn.edu>
Last modified: Mon Nov 18 13:32:34 1996