18.03 Class 35
, May 3, 2010
Linear Phase Portraits: Eigenvalues Rule
(usually)
1. Eigenvalues rule!
2. Tracedeterminant plane
3. Marginal cases
4. Stability
[1] Phase portrait: this means the
(x,y)
plane (the "phase plane")
with trajectories of solutions of
u' = Au
drawn on it (with direction of
time indicated).
These homogeneous linear equations exhibit a nice variety of phase portraits,
as shown by the Linear Phase Portraits Mathlets. An important fact is this:
EIGENVALUES RULE
We'll classify the linear phase portraits according to the eigenvalues
of the matrix A .
Example: those rabbits again:
A = [ .3 .1 ; .2 .4 ]
p_A(lambda) = lambda^2  .7 lambda + .1
has roots
lambda_1 = .5 ,
lambda_2 = .2
so we learn that all solutions flee from the origin: the phase portrait
is a "source."
lambda = .5 : [ .2 .1 ; .2 .1 ]
==>
v_1 = [ 1 ; 2 ]
so one normal mode is
u_1 = e^{.5 t} [ 1 ; 2 ]
If the number of rabbits in MacGregor's field is twice the number in
Jones's, it stays that way forever after.
lambda = .2 : [ .1 .1 ; .2 .2 ]
==>
v_2 [ 1 ; 1 ]
so the other normal mode is
u_2 = e^{.2t} [ 1 ; 1 ] .
This is not meaningful in itself in our population model,
but we can draw it in the phase plane. And the two together provide the
general solution.
What to the other trajectories look like?
For large
t ,
the
v_1
component is much bigger than the
v_2
component.
For small
t ,
the
v_1
component is much smaller than the v_2
component.
So near the origin the trajectories become asymototic to the eigenline with
smaller eigenvalue.
This phase portrait is a NODE. The same kind of picture occurs whenever the
eigenvalues are real, of the same sign, and distinct.
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[2] When are the eigenvalues nonreal?
[Slide:]
p_A(lambda) = det ( A  lambda I )
If
A = [ a b ; c d ]
then
p_A(lambda) = lambda^2  (tr A) lambda + (det A)
tr(A) = a + d = lambda_1 + lambda_2
det(A) = adbc = lambda_1 lambda)2
To find the eigenvalues, complete the square:
p_A(lambda) = (lambda  (tr(A)/2))^2 + (det(A)  (tr(A)/2)^2)
so
lambda1,2
=
tr(A)/2 + sqrt(tr(A)^2/4  det(A)).
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 Spring '09
 vogan
 Determinant, Addition, Matrices, Eigenvalue, eigenvector and eigenspace, Fundamental physics concepts, Normal mode

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