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CDS 140a Final Examination
J. Marsden, December, 2008
Attempt SEVEN of the following ten questions.
Each question is worth 20 points.
The exam time limit is three hours; no aids are permitted
.
Turn in the exam by 5pm on Thursday, December 11, 2008
.
Print Your Name
:
The SEVEN questions to be graded
:
1
2
3
4
5
6
7
8
9
10
/140
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2
1. Let
V
be a one dimensional potential whose derivative is given by
V
0
(
x
) = (
x

1)(
x

2)(
x

3)
and consider the system
˙
x
=
v
˙
v
=

V
0
(
x
)

νv
(a) Show that solutions of this system for any initial conditions exist for all
time.
(b) For
ν
= 0 argue that the system has both homoclinic and periodic orbits.
(c) What does Liapunov theory or La Salle’s invariance principle say about
the fate of solutions with
ν >
0?
2. Consider the following system in
R
2
:
˙
x
=
x
+
y
2
˙
y
=

y.
(a) Determine the stable and unstable subspaces of the linearization at the
origin.
(b) Calculate the explicit solution (
x
(
t
)
, y
(
t
)) of this system.
(c) Find an explicit expression for the stable and unstable manifolds of this
system.
3. Consider a system of the form
˙
x
=
f
1
(
x, y
)
˙
y
=
f
2
(
x, y
)
where
f
1
(0
,
0) =
f
2
(0
,
0) = 0 and
Df
(0) =
0

1
1
0
. Consider a candidate
Lyapunov function
V
(
x, y
) =
x
2
+
y
2
.
(a) Assume that
f
1
(
x, y
) and
f
2
(
x, y
) are polynomials in
x
and
y
of degree at
most two. Write
˙
V
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 Fall '09
 Marsden
 Stability theory, periodic orbits, rigid bar, J. Marsden

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