Page 357 - Advanced engineering mathematics

10.6 Phase Portraits 337

Case 5: A Has Pure Imaginary Eigenvalues
Now trajectories have the form
X(t) = c 1 [Ucos(βt) − Vsin(βt)]+ c 2 [Usin(βt) + Vcos(βt)].
Without an exponential factor to increase or decrease the length of this vector, trajectories now
are closed curves about the origin, representing a periodic solution. Now the origin is called a
center of the system.
In general, closed trajectories of a system represent periodic solutions.

EXAMPLE 10.24

Let

3 18
A = .
−1 −3
Eigenvalues of A are ±3i and eigenvectors are U ± iV, where

−3 −3
U = and V = .
1 0
Figure 10.24 is a phase portrait, showing closed trajectories moving (in this case) clockwise
about the origin, which is a center.

We now have a complete description of the trajectories of the real, linear 2 × 2system
X = AX. The qualitative behavior of the trajectories is determined by the eigenvalues, and we

have the following classification of the origin:

• Real, distinct eigenvalues of the same sign—(0,0) is a nodal source (positive eigenvalues) or
sink (negative eigenvalues);
• Real, distinct eigenvalues of the same sign—(0,0) is a saddle point;
• Equal eigenvalues, linearly independent eigenvectors—(0,0) is a proper node;
• Equal eigenvalues, all eigenvectors a multiple of a single eigenvector—(0,0) is an improper
node;

y

4

2

x
–15 –10 –5 0 5 10 15

–2

–4

FIGURE 10.24 Center in Example 10.24.

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October 14, 2010 20:32 THM/NEIL Page-337 27410_10_ch10_p295-342