Applications of Systems of Equations 42342y 0-2-4-4 -2 0 2 4
x
Figure 10 .5 Saddle Point
If a particle starts at a point on line m 1 in the fourth qu adrant, then c 2 = 0and c 1 > 0 , since x > 0 in the fourth quadrant. Since c 2 = 0 , the particle
remains on t he line m 1. And si n ce >. 1 = -3 < 0, the particle m oves toward
the origin as t ---+ oo. If a particle starts on line m 1 in the second quadrant,
then c 2 = 0 and c 1 < 0, since x < 0 in the second quadrant. Since c 2 = 0 , the
particle remains on the line m1 and since >-1 = -3 < 0, the particle moves
toward the origin as t ---+ oo. Similarly a particle which st arts at a point
on line m 2 remains on m 2 , but since >. 2 = 2 > 0 , the particle moves away
from t he origin as t ---+ oo. Next, suppose a particle starts at some point
not on line m 1 or m2- t hat is, suppose c 1 -=f. 0 and c2 -=f. 0. As t ---+ oo,
x(t) = c 1 e-^3 t + c2e^2 t ---+ c2e^2 t and y( t) = -4c 1 e-^3 t + c2e^2 t ---+ c2e^2 t. Thus, as
t ---+ oo,
G~~?) ---+ G~:~:) = c2e
2t G)
asymptotically. That is, any particle which starts at a point not on line m 1 or
m 2 approaches the line m2 asymptotically as t ---+ oo. Summarizing, we h ave
found that a particle which starts on the line m 1 moves toward the origin as
t ---+ oo, while a particle which does not start on m 1 ultimately moves away
from the origin and approaches the line m2. In this example, the origin is
an unstable crit ical point. The phase-plane portrait shown in Figure 10 .5
is typical for autonomous systems for which the eigenvalues are real and of
opposite sign. In such cases, the critical point (the origin) is called a saddle
point.