Page 351 - Advanced engineering mathematics

10.6 Phase Portraits 331

y
y
L 1
L 2
L
E 1 P 0 1
E 2 T 3
P 0
x
L 2
P 0 T 1
T 2
x

FIGURE 10.14 Eigenvectors E 1 , E 2 in
Case 1. FIGURE 10.15 Trajectories in Case 1-a.

3. If P 0 is on neither L 1 or L 2 , then the trajectory is a curve through P 0 having the parametric
form
λt
μt
X(t) = c 1 E 1 e + c 2 E 2 e .
Write this as
μt
X(t) = e [c 1 E 1 e (λ−μ)t + c 2 E 2 ].
Because λ − μ< 0, e (λ−μ)t → 0as t →∞ and the term c 1 E 1 e (λ−μ)t exerts increasingly less
influence on X(t). The trajectory still approaches the origin, but also approaches the line L 2
asymptotically as t →∞, as with T 3 in Figure 10.15.
A phase portrait of X =AX in this case therefore has all trajectories approaching the origin,

some along L 1 , some along L 2 , and all others asymptotic to L 2 . In this case, the origin is called
a nodal sink of the system. We can think of particles flowing along the trajectories toward (but
never quite reaching) the origin.

EXAMPLE 10.19
Suppose

−6 −2
A = .
5 1
A has eigenvalues and eigenvectors

2 −1
−1, and − 4, .
−5 1
Here λ =−4 and μ =−1. The general solution is

−1 −4t 2 −t
X(t) = c 1 e + c 2 e .
1 −5
L 1 is the line through the origin and (−1,1) and L 2 the line through the origin and (2,−5).
Figure 10.16 shows a phase portrait for this system. The origin is a nodal sink.

Case 1(b): The Eigenvalues are Positive, say 0 <μ<λ
Now the trajectories are the same as in Case 1 (a), but the flow is reversed. Instead of flowing
λt
into the origin, the trajectories are directed out of and away from the origin, because now e and
μt
e approach ∞ instead of zero as t →∞. All of the arrows on the trajectories now point away
from the origin and (0,0) is called a nodal source.

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October 14, 2010 20:32 THM/NEIL Page-331 27410_10_ch10_p295-342