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

Boyce W.E. Elementary Differential Equations and Boundary Value Problems - John Wiley & Sons, 2001. - 1310 p.
Download (direct link): elementarydifferentialequations2001.pdf
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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 i, where X and are real, X = 0, and > 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 i are typified by
*=( X ^ x (11)
4 x
or, in scalar form,
X1 = Xxj + 2,
X2 = .1 + 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 0 is the value of 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 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 < 0 and r to if > 0. Thus the trajectories are spirals, which approach or recede from the origin depending on the sign of . 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 ii, where 0, the trajectories are always spirals. They are directed inward or outward, respectively, depending on whether 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 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 < 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 r2/xf, we have
' \i.
(16)
Hence
it + 0,
(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 = + ip, r2 = ip. (a) < 0, the phase plane.
(b) < 0, xj versus t. (c) > 0, the phase plane. (d) > 0, xj versus t.
FIGURE 9.1.6 A spiral point; r = ip with < 0.
Pure Imaginary Eigenvalues. In this case = 0 and the system (11) reduces to
x = ' P P) x (19)
9.1 The Phase Plane: Linear Systems
467
with eigenvalues i. Using the same argument as in Case 4, we find that
r' = 0, ' = -i, (20)
and consequently,
r = c, = \it + 0, (21)
where c and 0 are constants. Thus the trajectories are circles, with center at the origin, that are traversed clockwise if > 0 and counterclockwise if < 0. A complete circuit about the origin is made in a time interval of length 2/, so all solutions are periodic with period 2/. The critical point is called a center.
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