This preview shows page 1. Sign up to view the full content.
Homework assignment 3
∗
Due date: Monday April 23, 2007.
1. Find the matrix exponentials of the following matrices (Don’t use software for your calcula
tions! You can use only use it to verify your results.):
p
0
1
−
1
0
P
,
0
−
1
−
1
−
1
0
−
1
−
1
−
1
0
,
1
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
,
1
1
0
0
0
0
1
1
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
,
1
1
0
0
0
0
1
0
0
0
0
0
1
0
0
0
0
0
1
1
0
0
0
0
1
.
2. Give an example of two 2
×
2 matrices
A
and
B
such that e
A
+
B
n
= e
A
e
B
.
3. Determine stability of the zero solution of the following equations:
•
¨
x
+ ˙
x
+ sin
x
= 0
.
•
¨
x
+ ˙
x
cos
x
+ sin
x
= 0
.
Note that these are problems 12
.
5 # 13 and # 14 from our text. Ignore the hint given there,
and instead try the following function:
V
(
x,y
) = 2 sin
2
±
x
2
²
+
y
2
2
.
4. Show that the competition model
˙
x
=
x
(
−
ax
−
by
+
r
1
)
˙
y
=
y
(
−
cx
−
dy
+
r
2
)
,
where all parameters
a,b,c,d,r
1
and
r
2
are positive, does not have a nonconstant periodic
solution in the region
D
:
D
=
{
(
x,y
)

x >
0
,y >
0
}
.
Hint: Apply Dulac’s criterion with appropriately chosen positive function
This is the end of the preview. Sign up
to
access the rest of the document.
 Summer '06
 DeLeenheer

Click to edit the document details