Lecture Notes in Differential Equations - Bruce E. Shapiro

383
eigendirection {λ 1 , v 1 }dominates and the solutions will approach lines parallel
to v 1 , while for large negative time (t → −∞) the solutions approach
the origin parallel to the slow eigendirection. The situation is reversed for
negative eigenvalues: the trajectories approach the origin along the slow
eigendirection as t → ∞ and diverge parallel to the fast eigendirection as
t → −∞.
Figure 34.4: Phase portrait for the system (34.56).
3
0
3
3 0 3
Example 34.3. Classify the fixed points and sketch the phase portrait of
the system
The matrix of coefficients is
}
x ′ = x + y
y ′ = −2x + 4y.
A =
( 1
) 1
−2 4
so that T = 5 and ∆ = 6. Consequently the eigenvalues are
(34.56)
(34.57)
λ = 1 (
T ± √ )
T
2
2 − 4∆
(34.58)
= 1 ( √ )
5 ± 25 − 24 (34.59)
2
= 3, 2 (34.60)