1550078481-Ordinary_Differential_Equations__Roberts_

Applications of Systems of Equations

EXAMPLE 5 An Asymptotically Unstable Critical Point

at the Origin

The eigenvalues of the system

x' - x + 6y

y' -3x + 5y

425

are .A= 2±3i. Since a = 2 > 0, the origin is an unstable critical point and the

trajectories spiral outward away from the origin in a clockwise direction. The
phase-plane portrait is the same as shown in Figure 10.6 except the direction
arrows on the trajectories must be reversed.

EXAMPLE 6 A Neutrally Stable Critical Point at the Origin

The eigenvalues of the system

x' x + 2y

y' -5x - y

are .A= ± 3i. Since a = 0, t he origin is a neutrally stable critical point. In
this case, the origin is called a center. The trajectories are "skewed ellipses"
with centers at the origin. A phase-plane portrait for this system is displayed
in Figure 10. 7.

4

2

y 0

-2

-4

-4 -2 0 2 4

x

Figure 10. 7 Neutrally Stable Center