1395.3 LOVE AFFAIRS
If % 0, at least one of the e igenvalues is zero. Then the or igin is not an isolate d
fixed point. There is either a whole line of fixed points, as in Figure5.1.5d, or a
plane of fixed points, if A 0.
Figure5.2.8 shows that saddle points, nodes, and spirals are the major types of
fixed points; they occur in large open regions of the ( %,U) plane. Centers, stars,
degenerate nodes, and non-isolated fixed points are borderline cases that occur
along curves in the ( %,U) plane. Of these borderline cases, centers are by far the
most important. They occur very commonly in frictionless mechanical systems
where energy is conserved.
EXAMPLE 5.2.6:
Classify the fixed point x* 0 for the system
xxA, where A=⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
12
34
.
Solution: The matrix has % 2; hence the fixed point is a saddle point. ■
EXAMPLE 5.2.7:
Redo Example 5.2.6 for A=⎛
⎝
⎜
⎜
⎜
⎜
⎞
⎠
⎟
⎟
⎟
⎟
21
34
.
Solution: Now % 5 and U 6. Since % 0 and U
2 4% 16 0, the fixed
point is a node. It is unstable, since U 0. ■
5.3 Love Affairs
To arouse your interest in the classification of linear systems, we now discuss a
simple model for the dynamics of love affairs (Strogatz 1988). The following story
illustrates the idea.
Romeo is in love with Juliet, but in our version of this story, Juliet is a fickle
lover. The more Romeo loves her, the more Juliet wants to run away and hide. But
when Romeo gets discouraged and backs off, Juliet begins to find him strangely
attractive. Romeo, on the other hand, tends to echo her: he warms up when she
loves him, and grows cold when she hates him.
Let
R ( t ) Romeo’s love / hate for Juliet at time t
J ( t ) Juliet’s love / hate for Romeo at time t.
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