© 2008
Zachary S Tseng
D
-2 - 28
Exercises D-2.2
:
Find all the critical point(s) of each nonlinear system given.
Then determine
the type and stability of each critical point.
1.
x
′ =
xy
+ 3
y
y
′ =
xy
− 3
x
2.
x
′ =
x
2
− 3
xy
+ 2
x
y
′ =
x
+
y
− 1
3.
x
′ =
x
2
+
y
2
− 13
y
′ =
xy
− 2
x
− 2
y
+ 4
4.
x
′ =
2 −
x
2
−
y
2
y
′ =
x
2
−
y
2
5.
x
′ =
x
2
y
+ 3
xy
– 10
y
y
′ =
xy
− 4
x
Answer D-2.2
:
1.
Critical points are (0, 0) and (−3, 3).
(0, 0) is a stable center, and (−3, 3)
is an unstable saddle point.
2.
Critical points are (0, 1) and (1/4, 3/4).
(0, 1) is an unstable saddle point,
and (1/4, 3/4) is an unstable spiral point.
3.
Critical points are (2, 3), (2, −3), (3, 2), and (−3, 2).
(2, 3) is an unstable
saddle point, (2, −3) is an unstable saddle point, (3, 2) is an unstable node,
and (−3, 2) is an asymptotically stable node.
4.
Critical points are (1, 1), (1, −1), (−1, 1), and (−1, −1).
(1, 1) is an
asymptotically stable spiral point, (1, −1) and (−1, 1) both are unstable
saddle points, and (−1, −1) is an unstable spiral point.
5. Critical points are (0, 0), (2, 4), and (−5, 4).
(0, 0) is an unstable saddle
point, (2, 4) is an unstable node, and (−5, 4) is an asymptotically stable node.