April 27, 2006
75 minutes
L. Healy
1. [20 points] For each of the following sets of diﬀerential equations with
dynamical variables
x
,
y
,
z
, determine
❶
whether the equations divide
into sets of equations that are decoupled from each other or if all three
equations must be considered as one set,
❷
for each of the sets, whether
the equations are linear or nonlinear,
❸
if there is a oneequation linear
set, under what condition it is stable or unstable,
❹
if there is a stable
oneequation linear set, what its frequency of oscillation is.
(a)
˙
x
+
αyz
=0
˙
y
+
βxz
=0
˙
z
+
γxy
=0
❖
Nonlinear, doesn’t decouple
(b)
˙
x
+
αz
=0
˙
y
+
βy
=0
˙
z
+
γx
=0
❖
Linear;
y
decouples from
x

z
. Motion in
y
is stable if
β
is
imaginary, frequecy is

β

.
(c)
˙
x
+
αx
=0
˙
y
+
βy
2
=0
˙
z
+
γyz
=0
❖
Motion in
x
is linear and decoupled from others. Motion in
y
is nonlinear and decoupled from others. Motion in
z
is nonlinear
and coupled to
y
. Motion in
x
is stable if real part of
α
is positive
<
(
α
)
>
0, frequency is imaginary part
=
(
α
).
1
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