.
(Affine just means ``linear plus a constant term''.)
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
as a dissipation parameter; we will
reserve D to mean the ``Jacobian operator.'')
, 
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.
. Clear all trajectories.
For EACH of the two equilibria you found:
values for which 
and 
, i=0,1. on the printout.)
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.
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.)
, 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:
into pplane. (You may need to add a constant to
the equations that define the linear vector field.) .Question: Does the affine vector field approximate the pendulum field near equilibria? How well? How could you make the approximation better?
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!
. 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?
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.