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10.6 Phase Portraits 335

y

L
c W + c E + c Et
1
1
2
X(t)
W
E
x

FIGURE 10.21 Trajectory formed from W
and E in Case 3(b).

The origin in this case is called an improper node of X = AX. The next example shows

typical trajectories in the case of an improper node.

EXAMPLE 10.22
Let

−10 6
A = .
−6 2
A has an eigenvalue of −4, and every eigenvector is a nonzero multiple of

1
E = .
1
A routine calculation gives
1

W = .
7/6
The general solution is

t + 1 −4t 1 −4t
X(t) = c 1 e + c 2 e .
t + 7/6 1
Figure 10.22 is a phase portrait for this system. The trajectories approach the origin tangent to
the line through E, when this vector is represented as an arrow from the origin. The origin is an
improper node for this system.

Case 4: A Has Complex Eigenvalues With Nonzero Real Part
Let λ = α + iβ be an eigenvalue with α = 0 and eigenvector U + iV. Then the general solution is
αt
αt
X(t) = c 1 e [Ucos(βt) − vsin(βt)]+ c 2 e [Usin(βt) + Vsin(βt)].
The trigonometric terms cause the solution vector X(t) to rotate as t increases, while if α< 0,
the length of X(t) decreases to zero. Thus, trajectories spiral inward toward the origin as t →∞
and the origin is called a spiral sink.
If α>0, the trajectories spiral outward from the origin as t increases, and the origin is called
a spiral source.
In both cases, we call the origin a spiral point.

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October 14, 2010 20:32 THM/NEIL Page-335 27410_10_ch10_p295-342