Lecture Notes in Differential Equations - Bruce E. Shapiro

384 LESSON 34. CRITICAL POINTS
The eigenvalues are real, positive, and distinct, so the fixed point is a
source. ( The fast eigenvector, corresponding to the larger eigenvalue, λ = 3,
1
is v = . The slow eigenvector, corresponding to the smaller eigenvalue
2)
( 1
λ = 2, is w = . Near the origin, the slow eigendirection dominates,
1)
while further away, the fast eigendirection dominates. The trajectories
mostly leave the origin tangentially to the line y = x, which is along the
slow eigendirection, then bend around parallel the fast eigendirection as
one moves away from the origin. The phase portrait is illustrated in figure
34.4.
Repeated Real Nonzero Eigenvalues
If T 2 = 4∆ ≠ 0, the eigenvalues are real and repeated (we exclude the case
with both eigenvalues equal to zero for now because that only occurs when
the matrix of coefficients has a determinant of zero). In this case we are
not required to have two linearly independent eigenvectors.
If there are two linearly independent eigenvectors v and w, then the solution
is
y = (Av + Btw)e λt (34.61)
and if there is only a single eigenvector v then
y = [Av + B(tv + w)]e λt (34.62)
where w is the generalized eigenvector satisfying (A − λI)w = v.
In the first case (linearly independent eigenvectors) all solutions lie on
straight lines passing through the origin, approaching the origin in positive
time (t → ∞) if λ > 0, and in negative time (t → −∞) if λ < 0.
The critical point at the origin is called a star node: a stable star if
λ = T/2 < 0 and an unstable star if λ = T/2 > 0.
If there is only a single eigenvector then the trajectories approach the origin
tangent to v; as one moves away from the origin, the trajectories bend
around and diverge parallel to v (see figure 34.5). The origin is called either
a stable degenerate node (λ = T/2 < 0) or an unstable degenerate
node (λ = T/2 > 0).