# Elementary Differential Equations and Boundary Value Problems - Boyce W.E.

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464

Chapter 9. Nonlinear Differential Equations and Stability

xv

.

x2 ^ ^ ^Increasing.

C2^^\

Increasing t^^^ \

A t X

C2 < 0 /

Cj\ + C2^^>"'

'c2 > 0

(a) ()

Y Increasing .

0 C2^ „

> 1. ,

>2

C2

c2 < 0 V^Cj^ + C2^ Xj

Increasing t <C2,

(c)

FIGURE 9.1.4 An improper node, one independent eigenvector; rx = r2 < 0. (a) The phase plane. (b) xj versus t. (c) The phase plane.

CASE 4 Complex Eigenvalues. Suppose that the eigenvalues are X ± iã, where X and ã are real, X = 0, and ã > 0. It is possible to write down the general solution in terms of the eigenvalues and eigenvectors, as shown in Section 7.6. However, we proceed in a different way.

Systems having the eigenvalues X ± iã are typified by

*=( X ^ x (11)

4—ã x

or, in scalar form,

X1 = Xxj + ãõ2,

X2 = —ö.õ1 + Xx2

(12)

9.1 The Phase Plane: Linear Systems

465

(13)

(14)

(15)

where c is a constant. Similarly, substituting from Eqs. (12) in the second of Eqs. (13),

where â0 is the value of â when t — 0.

Equations (15) and (17) are parametric equations in polar coordinates of the trajectories of the system (11). Since i > 0, it follows from Eq. (17) that â decreases as t increases, so the direction of motion on a trajectory is clockwise. As t — to, we see from Eq. (15) that r — 0 if ê < 0 and r —to if ê > 0. Thus the trajectories are spirals, which approach or recede from the origin depending on the sign of ê. Both possibilities are shown in Figure 9.1.5, along with some typical graphs of x1 versus t. The critical point is called a spiral point in this case. Frequently, the terms spiral sink and spiral source, respectively, are used to refer to spiral points whose trajectories approach, or depart from, the critical point.

More generally, it is possible to show that for any system with complex eigenvalues ê ± ii, where ê — 0, the trajectories are always spirals. They are directed inward or outward, respectively, depending on whether ê is negative or positive. They may be elongated and skewed with respect to the coordinate axes, and the direction of motion may be either clockwise or counterclockwise. While a detailed analysis is moderately difficult, it is easy to obtain a general idea of the orientation of the trajectories directly from the differential equations. Suppose that

has complex eigenvalues ê ± ii, and look at the point (0, 1) on the positive /-axis. At this point it follows from Eqs. (18) that dx/dt — b and dy/dt — d. Depending on the signs of b and d, one can infer the direction of motion and the approximate orientation of the trajectories. For instance, if both b and d are negative, then the trajectories cross the positive y-axis so as to move down and into the second quadrant. If ê < 0 also, then the trajectories must be inward-pointing spirals resembling the one in Figure 9.1.6. Another case was given in Example 1 of Section 7.6, whose trajectories are shown in Figure 7.6.2.

and using the fact that sec2 â — r2/xf, we have

â' — —\i.

(16)

Hence

â — —it + â0,

(17)

(18)

466

Chapter 9. Nonlinear Differential Equations and Stability

CASE 5

x 1

,

1

x _ /

x2'

((fa

(c) (d

FIGURE 9.1.5 A spiral point; rx = ê + ip, r2 = ê — ip. (a) ê< 0, the phase plane.

(b) ê < 0, xj versus t. (c) ê > 0, the phase plane. (d) ê > 0, xj versus t.

FIGURE 9.1.6 A spiral point; r = ê ± ip with ê < 0.

Pure Imaginary Eigenvalues. In this case ê = 0 and the system (11) reduces to

x = ' —P P) x (19)

9.1 The Phase Plane: Linear Systems

467

with eigenvalues ±ië. Using the same argument as in Case 4, we find that

r' = 0, â' = -i, (20)

and consequently,

r = c, â = —\it + â0, (21)

where c and â0 are constants. Thus the trajectories are circles, with center at the origin, that are traversed clockwise if ë > 0 and counterclockwise if ë < 0. A complete circuit about the origin is made in a time interval of length 2ï/ë, so all solutions are periodic with period 2ï/ë. The critical point is called a center.

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