Calc III Lab: Saddles
The phase portraits you will see in this section of the lab are called
saddles. They are characterized by having one direction in
which trajectories are attracted to the equilibrium, and another
direction in which trajectories move away from the equilibrium.
Activity
- Compute solutions to the linear differential equation determined by
the matrix with entries
(a,b,c,d)=(1,-1,0.5,-1). Enter the usual
information on the supplied diagram.
- Do the same for the linear differential equation determined by
the matrix with entries
(a,b,c,d)=(1,1,0.5,0).
Question 2
- Describe a physical or biological situation in which can be modeled by
a differential equation whose phase portrait is a saddle.
Specify the quantities that are
evolving in time, and indicate how "unstableness" of the
phase portrait corresponds to your quantities.
- Compare and contrast the "node" phase portrait with the saddle
and the foci. In what ways are they similar and different?
Question 3
The trace and determinant almost (but not
quite!) completely determine the dynamics of a linear differential
equation. Choose one of the matrices that we investigated. Find a
different matrix that has the same
trace and determinant but which has different dynamics. For example,
one the phase portrait might wind in the clockwise direction, whereas
the other winds counterclockwise.
Go To
Robert E. Thurman <thurman@geom.umn.edu>
Last modified: Mon Nov 18 13:32:35 1996