Lab #8

This worksheet does not need to be handed in, but you are responsible for the material. The purpose of the lab is to understand how to completely understand the qualitative phase portraits of linear vector fields. (Intuitively, we say that two differential equations are qualitatively similar if their phase portraits look the same.) As we saw in a previous lab, this means that we can understand the local behavior of most vector fields near equilibria.

In a previous lab we saw that if (p, q) is a ``typical'' equilibrium of a vector field defined by , then

In words, the equation above says that near a typical equilibrium, vector fields ``look'' like linear vector fields, so in this lab we will classify the appearance of most linear vector fields. For convenience, we will choose the equilibrium point to be at the origin, (p,q)=(0,0). Then the matrix is the coefficient matrix for the linear vector field

Recall that the determinant of the above matrix is

and the trace of the matrix is

To start this lab, launch Matlab and pplane. Choose the ``linear system'' example from pplane's gallery of pre-defined differential equations. (Note that pplane uses capital letters for its matrix parameters.)

ACTIVITY 1: Centers The phase portraits you will see in this section of the lab are called centers. Change the values of the coefficient matrix to (a,b,c,d)= (0,1,-0.5,0) so that we may study the phase portrait of the differential equation

1. Compute a phase portrait for the linear system given above.
2. Compute the trace and determinant of the matrix M for the specific values of coefficients (a,b,c,d) used above.
3. Plot the ordered pair on the diagram provided.
4. Next to the ordered pair you just plotted, sketch the phase portrait of the linear ODE associated with this matrix. [This has been done for you!] Put arrows on the phase portrait to indicate the direction that trajectories flow.
5. Clear the old phase portrait. Change the entries of M to (a,b,c,d)=(0,1,-0.125,0). Repeat the steps specified above to plot the trace and determinant on the supplied diagram, and to sketch the phase portrait of the associated linear ODE.

ACTIVITY 2: 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).

1. Clear the old phase portrait. Compute a phase portrait to the linear differential equation determined by the matrix with entries (a,b,c,d)=(-1,1,-0.5,0).
2. Compute the trace and determinant of the associated matrix M.
3. Again, plot the ordered pair on the diagram provided. Sketch the phase portrait on the diagram and put arrows on the phase portrait to indicate the direction that trajectories flow.
4. Clear the old phase portrait and do the same activity for the linear differential equation determined by the matrix with entries (a,b,c,d)=(1,1,-0.5,0).

ACTIVITY 3: Nodes The phase portraits you will see in this section of the lab are called nodes. As was the case for the focus equilibria, the equilibrium in the phase portrait for a node is referred to as a ``sink'' or a ``source,'' depending on whether or not it is locally attracting.

1. Clear the old phase portrait. Compute the phase portrait for the linear differential equation determined by the matrix with entries (a,b,c,d)=(-1,1,-0.125,0).
2. Compute the trace and determinant of the associated matrix.
3. Plot the ordered pair on the diagram provided, and sketch the phase portrait (with arrows!) on the diagram.
4. Do the same for the linear differential equation determined by the matrix with entries (a,b,c,d)=(1,1,-0.125,0).

ACTIVITY 4: 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.

1. Compute solutions to the linear differential equation determined by the matrix with entries (a,b,c,d)=(1,1,0.5,0). Enter the usual information on the supplied diagram.
2. Do the same for the linear differential equation determined by the matrix with entries .

ACTIVITY 5: In your group, choose three random matrices. Compute their trace and determinant. Using your previous work, predict what the phase portrait will look like before you compute it.

ACTIVITY 6: 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.

ACTIVITY 7: Create a coefficient matrix and plot the linear phase portrait for a linear system whose coefficient matrix satisfies
1. ,
2. ,

In what ways might these phase portraits be considered ``degenerate'' or ``atypical''? (Hint: For the last phase portrait, think about the way phase portraits ``evolve'' from a foci to a node.)

Figure:   Classifying linear phase portraits according to their determinant and trace.