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\\/2sin\/2 ó \-ë/2cosë/2 ó
The real and imaginary parts of this solution form a pair of linearly independent real-valued solutions of Eq. (19):
/a -t ( cosV21 \ -t ( sin/21 \
u(t) = " Usi,,V2 t) ¦ v(t) = " i-^cos^ J ¦ (25)
Hence the general solution of Eqs. (19) is
I^ = V ' ( J?s ^ ) + c2e-' ( . (26)
V) 1 \V2sin\/2 t/ 2 \-ë/2cosë/21
Upon imposing the initial conditions
VI (0) = 22 ¦ (27)
we find that
÷0) + cA -v°) = (2
Thus cx = 2 and c2 = -ë/2. The solution of the stated problem is then given by Eq. (26) with these values of cx and c2. The graph of the solution is shown in Figure
7.6.4. The trajectory spirals counterclockwise and rapidly approaches the origin, due to the factor e-t.
7.6 Complex Eigenvalues
FIGURE 7.6.4 Solution of the initial value problem in Example 2.
x =< —2 2) x (29)
contains a parameter à. Describe how the solutions depend qualitatively on a; in particular, find the critical values of a at which the qualitative behavior of the trajectories in the phase plane changes markedly.
The behavior of the trajectories is controlled by the eigenvalues of the coefficient matrix. The characteristic equation is
r2 — ar + 4 = 0, (30)
so the eigenvalues are
a ± Va2 — 16
r = ^-----------------• (31)
From Eq. (31) it follows that the eigenvalues are complex conjugates for —4 < a < 4 and are real otherwise. Thus two critical values are a = —4 and a = 4, where the eigenvalues change from real to complex, or vice versa. For a < —4 both eigenvalues are negative, so all trajectories approach the origin, which is an asymptotically stable node. For a > 4 both eigenvalues are positive, so the origin is again a node, this time unstable; all trajectories (except x = 0) become unbounded. In the intermediate range, —4 < a < 4, the eigenvalues are complex and the trajectories are spirals. However, for —4 < a < 0 the real part of the eigenvalues is negative, the spirals are directed inward, and the origin is asymptotically stable, while for 0 < a < 4 the opposite is the case and the origin is unstable. Thus a = 0 is also a critical value where the direction of the spirals changes from inward to outward. For this value of a the origin is a center and the trajectories are closed curves about the origin, corresponding to solutions that are periodic in time. The other critical values, a = ±4, yield eigenvalues that are real and equal. In this case the origin is again a node, but the phase portrait differs somewhat from those in Section 7.5. We take up this case in Section 7.8.
For second order systems with real coefficients we have now completed our description of the three main cases that can occur.
1. Eigenvalues have opposite signs; x = 0 is a saddle point.
Chapter 7. Systems of First Order Linear Equations
2. Eigenvalues have the same sign but are unequal; x = 0 is a node.
3. Eigenvalues are complex with nonzero real part; x = 0 is a spiral point.
Other possibilities are of less importance and occur as transitions between two of the cases just listed. For example, a zero eigenvalue occurs during the transition between a saddle point and a node. Purely imaginary eigenvalues occur during a transition between asymptotically stable and unstable spiral points. Finally, real and equal eigenvalues appear during the transition between nodes and spiral points.
In each of Problems 1 through 8 express the general solution of the given system of equations in terms of real-valued functions. In each of Problems 1 through 6 also draw a direction field, sketch a few of the trajectories, and describe the behavior of the solutions as t ^ro.
1. x' = x > 2. x' =
3. x' = x > 4. x' = I n 2 I x
5. x' = ( _ „ I x > 6. x' =
' 1 2
0 0\ /-3 0
7. x = I 2 1 _2 I x 8. x' = I 1 _1
2 1/ \_2 _1
In each of Problems 9 and 10 find the solution of the given initial value problem. Describe the behavior of the solution as t .