1550078481-Ordinary_Differential_Equations__Roberts_

424 Ordinary Differential Equations

Now suppose that .A = a + (3i where (3 #- 0 is an eigenvalue of the real
matrix A and w = u +iv is an associated eigenvector. We saw in chapter 8

that the general solution of the system ( 4) dz/ dt = Az can be written as

z = c 1 { ( e°'t cos (3t )u - ( e°'t sin (3t )v} + c 2 { ( e°'t sin (3t )u + ( e°'t cos (3t )v}

where c 1 and c 2 are arbitrary real constants. Recall that the first and sec-
ond components of z are x(t) and y(t), respectively. Since I sinf3tl :S 1,
I cosf3t1 :S 1, and u and v are real constant vectors, if a< 0, then as t----> oo,
x(t) ----> 0 and y(t) ----> 0 and the origin is an asymptotically stable critical
point. The functions sin (3t and cos (3t cause the trajectories to rotate about
the origin as they approach forming "spirals." If a > 0, the trajectories spi-
ral outward away from the origin and the origin is an unstable critical point.

When a = 0, the trajectories are elliptical in appearance with centers at the

origin. In this case, the origin is a neutrally stable critical point. It can be
shown that when the system (4) has complex conjugate eigenvalues .A= a±i(3,
the trajectories are always spirals when a#-0. The spirals may be elongated
and skewed with respect to the coordinate axes and may spiral clockwise or
counterclockwise about the origin.

EXAMPLE 4 An Asymptotically Stable Critical Point

at the Origin

The eigenvalues of the system

x' x - 6y

y' 3x - 5y

are .A = -2 ± 3i. Since a = -2 < 0, the ongm is an asymptotically
stable critical point and the trajectories spiral inward toward the origin in a
counterclockwise direction as shown in the phase-plane portrait of Figure 10 .6.

4

2

y 0

-2

-4

-4 -2 0 2 4

x

Figure 10.6 Asymptotically Stable Spiral Point