Professor Tse
Page 1 of 7
Spring 2007 Dynamic Systems
Midterm Solution
Problem 1. Short Answer Questions
(a) Model Matching
The fate of the Universe
“GrabberHolder Model”
Reason. The grabber is “expansion push of Big Bang,” and the holder is “the
gravitational attraction force.”
The Clubbing Scene
“Two Sided Market”
Reason. The number of men and women in the club depends on each other. If one of
them increases so does the other and vice versa.
Note: Partial credit has been given even if you selected another model and made an
appropriate explanation for it.
(b) True/False
(i) False.
First, a nonlinear system may not always have an equilibrium point.
Second, even if it has an equilibrium point, it may not be determined by
analyzing the stability of a linearized system. Think of the case when all
0
=
r
λ
for a continuous system and when
1


=
λ
for a discrete system.
(ii)
2
4
)
(
)
(
2
2
,
1
bc
d
a
d
a
−
−
±
+
=
λ

True.
bc
a
−
±
=
2
,
1
λ
; since
, we have a spiral out behavior.
0
>
a

False.
bc
−
±
=
2
,
1
λ
; since
0
=
r
λ
, we have a cycling behavior.

True.
bc
a
−
±
=
2
,
1
λ
; since
, we have a spiral in behavior.
0
<
a

False.
bc
a
−
±
=
2
,
1
λ
; since
, we have real eigenvalues.
0
>
−
bc
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