Lab #7

This worksheet does not need to be handed in, but you are responsible for the material. The purpose of the lab is to explore the idea that linear vector fields can approximate nonlinear vector fields, and that this approximation is particularly useful near an equilibrium point.

We know that if F is a differentiable function of two variables, then the function F is well-approximated near the point (p, q) by the affine function . (Affine just means ``linear plus a constant term''.)

Note that if is a vector field, and if (p,q) is an equilibrium, then F(p,q) is the zero vector (why!). Then near a ``typical'' equilibrium,

Remark: The word ``typical'' is used to eliminate cases where, for example, DF(p,q) might be the zero matrix. A ``non-typical'' equilibrium might be, for example, a critical point of a gradient vector field at which the second derivative test fails to classify the equilibrium as a min/max/saddle.

In words, the equation above says that near a typical equilibrium, vector fields ``look'' like linear vector fields! We have already seen in an earlier lab that AWAY from equilibria, vector fields look constant, so this lab will focus on approximations near equilibria. (To be honest, the vector fields we will consider in this lab are ``affine'' rather than linear, but they can be made linear by a simple change of coordinates, so we will sometimes continue to use the slightly inappropriate term ``linear'' to refer to vector fields that contain only first powers of the variables.)

In this lab we will study the pendulum differential equation

The variable measures the angle of the pendulum bob, using ``straight down'' to be . In particular, the pendulum is straight up when (or ). The variable v is an angular velocity. The parameter is a ``dissipation'' (or damping) parameter: gives us a Hamiltonian differential equation whose solutions run along level sets of the Hamiltonian function (a.k.a, ``the energy surface''), whereas is the physically realistic case in which the pendulum encounters resistance from the surrounding medium, and so loses energy.

To get started,

setenv PRINTER line

prior to launching Matlab and pplane. Choose the ``pendulum'' example from pplane's gallery of pre-defined differential equations. (Note that pplane uses D instead of as a dissipation parameter; we will reserve D to mean the ``Jacobian operator.'')

ACTIVITY 1: Set the display domain to , and generate a phase portrait for the pendulum equation with . There are two equilibria: one a center at , and the other a saddle at . Analytically locate the two equilibria (that is, find the values of and ) and compute the Jacobian of the vector field evaluated at each equilibrium. Write down the evaluated Jacobian for later reference.

ACTIVITY 2: Make sure . Clear all trajectories. For EACH of the two equilibria you found:
1. Use pplane to numerically locate the equilibrium.
2. Zoom in to a region of ``radius 1/2'' about the equilibrium. That is, the display window should show values for which and , i=0,1.
3. Generate a ``local phase portrait'' of the pendulum near each equilibrium. (Hint: For easy comparison with later phase portraits, try to choose initial conditions that are reproducible. For example, you might choose initial conditions spaced uniformly along the boundary of the display window.)
4. Print out this local phase portrait. (Hint: to easily distinguish your printouts, investigate the item under the PPLANE Options menu that enables you to ``Enter text on the Display Window.'' You will have to type the text into the shell window from which you launched matlab. Include the fact that on the printout.)

ACTIVITY 3: Choose a value of from the set . Reset the domain to and and generate a phase portrait for the pendulum equation, now with . Again, there are two equilibria: let the stable equilibrium be at . Analytically locate the position of the stable equilibrium and compute (and record) the Jacobian of the vector field evaluated at .

Repeat Activity 2 to generate a local phase portrait near . Include your value of on the printout.

At this point you have three local phase portraits. You also have three matrices that represent the affine vector fields that best approximate the vector field of the pendulum near each equilibrium. How do these matrices give rise to an affine vector field, and how well does that vector field approximate the phase portraits we've computed?

For a matrix , the matrix generates affine vector fields of the form

In practice, the constants (p,q) are often the position of an equilibrium, and the matrix is obtained by evaluating the Jacobian at that equilibrium.

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 4: For each pendulum equilibrium, , i=0,1,2, set the domain of the display window to the same domain values you used in Activities 3 and 4 when you generated local phase portraits of the pendulum.

For each i=0,1,2:

1. Input the coefficients of into pplane. (You may need to add a constant to the equations that define the linear vector field.)
2. Generate a phase portrait for the affine system defined by the matrix .
3. Print out the linear phase portrait and superimpose it onto the corresponding local phase portrait of the pendulum equation. (For example, place one sheet of paper over the other and hold them up to the light.)

Question: Does the affine vector field approximate the pendulum field near equilibria? How well? How could you make the approximation better?

Near a saddle equilibrium, there are trajectories that asymptotically approach/leave the equilibrium. These may be numerically found in pplane by locating the equilibria, and then choosing the item labeled ``Plot stable and unstable orbits'' under the PPLANE Options menu. Four distinct trajectories approach the equilibria, so that the phase portrait locally looks like an ``X''. We will call any of these four special trajectories a separatrix (the plural form is separatrices). In other words, a separatrix is a trajectory that asymptotically approaches a saddle equilibrium as or as .

Remark: if the vector field is linear, then the separatrices really do form an ``X,'' but as you will see from the pendulum equation, in general the separatrices may be curves that are not straight!

ACTIVITY 5: Return to the pendulum equation with . Compute the separatrices. Zoom in near the saddle until the separatrices look like straight lines, then graphically estimate the slope of the separatrices as they approach the saddle point.

Now switch to the linear vector field that best approximates the pendulum field near the saddle point. Find the separatrices of the linear vector field and estimate the slope of those curves.

What is the relationship between the slopes of the linear and nonlinear separatrices?

If we restrict ourselves to consider affine vector fields, then separatrices are the images of lines that approach the equilibrium. Geometrically, consider the vector, u, from the equilibrium to a point ON A SEPARATRIX near the equilibrium. In symbols, . Note that u points along the (straight) separatrix. Since is on a separatrix, the vector field at either points in the same direction as u, or else it points in the opposite direction.

Said another way, the vector u is a scalar multiple of the vector field at . In symbols, if , the vector field is just the matrix-vector product J u. If is an arbitrary scalar, then the requirement that u is a scalar multiple of the vector field, says that u satisfies the equation

This equation is extremely important in higher mathematics. Vectors u that satisfy the equation are called eigenvectors; the constant is called an eigenvalue. We will learn more about eigenvalues and eigenvectors next quarter.

ACTIVITY 6: On your printout of the phase portrait of the linear vector field with a saddle equilibrium, sketch the geometry described in the previous paragraphs.