Exercises for Qualitative Analysis Module
1.
Purpose:
To study nonlinear centers.
Notes:
Define a
reversible system
to be any secondorder system that is invariant under
t
→ 
t
and
y
→ 
y
. For example, any system ofthe form
˙
x
=
f
(
x, y
)
˙
y
=
g
(
x, y
)
,
where
f
is odd in
y
and
g
is even in
y
(i.e.,
f
(
x,

y
) =

f
(
x, y
) and
g
(
x,

y
) =
g
(
x, y
)) is reversible.
Theorem:
(Nonlinear centers for reversible systems) Suppose the origin
x
*
=
0
is a linear center for
the continously differentiable system
˙
x
=
f
(
x, y
)
˙
y
=
g
(
x, y
)
,
and suppose that the system is reversible. Then sufficiently close to the origin, all trajectories are
closed curves.
from[Str94], Example 6.6.1
Exercise:
Show that the system
˙
x
=
y

y
3
˙
y
=

x

y
2
has a nonlinear center at the origin, and plot the phase portrait.
Solution:
For the system
˙
x
=
y

y
3
=
f
(
x, y
)
˙
y
=

x

y
2
=
g
(
x, y
)
the fixed point
x
∗
is (0
,
0). Compute the Jacobian for the system and evaluate it at the fixed point,
A
=
bracketleftbigg
0
1

3
y
2

1

2
y
bracketrightbigg
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
(0
,
0)
=
bracketleftbigg
0
1

1
0
bracketrightbigg
.
The eigenvalues of
A
are
i
and

i
; purely imaginary eigenvalues are characteristic of a center; thus
x
∗
= 0 is a center. To show that the system is reversible, consider
f
(
x,

y
) =

y

(

y
)
3
=

y
+
y
3
=

(
y

y
3
) =

f
(
x, y
)
g
(
x,

y
) =

x

(

y
)
2
=

x

y
2
=
g
(
x, y
)
which illustrates that
f
is odd in
y
and
g
is even in
y
. Thus all trajectories sufficiently close to the
origin are closed curves. The phase portrait is shown in Figure 11.