For the case of no damping (b = 0) the eigenvalues are
k 1;2 ¼
Æix 0 for k even,
Æx 0 for k odd:
&
ð3:33Þ
For singular points of even k (h = …, –2p, 0, 2p, …) both eigenvalues are
imaginary. This is the critical case described above (max[Re(k j )] = 0). Comparing
with Fig. 3.6 we see that these singular points are neither stable nor unstable:
Nearby orbits stay near, but are neither attracted to, nor repelled from the singular
points corresponding to the down-pointing position of the pendulum. Singular
points with odd k (h = …, –p, p, 3p, …) are unstable, since one eigenvalue has a
positive real part (k = +x 0 ). This also appears from the orbits of Fig. 3.6.
For subcritical damping (0 < b < 1) we may write the Jacobian eigenvalues
(3.32) in the form
k 1;2 ¼
Àb Æ i
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b
2
p
x 0 for k even,
Àb Æ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ b
2
p
x 0 for k odd:
8
<
:
ð3:34Þ
Thus singular points of even k are stable, since both eigenvalues have negative
real parts. Singular points of odd k are unstable, since (1 + b
2 )
1/2 > b when
0 < b < 1, implying that one of the eigenvalues has a positive real part. These
results agree with the flow of orbits in Fig. 3.7.
3.4.5 On the Behavior of Orbits Near Singular Points
The above analysis of stability is based on Jacobian eigenvalues. These eigenvalues
merely reveal whether a singular point attracts or repels nearby orbits. To obtain a
more detailed picture on how orbits are disturbed in the vicinity of a singular point,
one needs to consider also the associated eigenvectors. The eigenvectors of the
linearized system will provide local information on orbits for the nonlinear system
as well, since near singular points the orbits of the nonlinear system are well
approximated by those of the linearized system.
Be careful to note that only the local behavior near singular points can be
assessed this way. To obtain a global orbital picture is kind of an art, requiring all
the local pictures to be tied together in the right manner; often this is not needed,
though.
Inspecting (3.27) it appears that the character of the linearized solutions depends
on whether the eigenvalues k are real or complex, and whether the eigenvalues are
distinct. Real eigenvalues cause the motion to grow or shrink exponentially in time,
whereas complex eigenvalues introduce oscillatory components in the motion
(since, with c, x 2 R: k = c + ix ) e
kt = e
c t (cos(xt) + i sin(xt))). Non-distinct
110
3 Nonlinear Vibrations: Classical Local Theory
k 1;2 ¼
Æix 0 for k even,
Æx 0 for k odd:
&
ð3:33Þ
For singular points of even k (h = …, –2p, 0, 2p, …) both eigenvalues are
imaginary. This is the critical case described above (max[Re(k j )] = 0). Comparing
with Fig. 3.6 we see that these singular points are neither stable nor unstable:
Nearby orbits stay near, but are neither attracted to, nor repelled from the singular
points corresponding to the down-pointing position of the pendulum. Singular
points with odd k (h = …, –p, p, 3p, …) are unstable, since one eigenvalue has a
positive real part (k = +x 0 ). This also appears from the orbits of Fig. 3.6.
For subcritical damping (0 < b < 1) we may write the Jacobian eigenvalues
(3.32) in the form
k 1;2 ¼
Àb Æ i
ffiffiffiffiffiffiffiffiffiffiffiffiffi
1 À b
2
p
x 0 for k even,
Àb Æ
ffiffiffiffiffiffiffiffiffiffiffiffi ffi
1 þ b
2
p
x 0 for k odd:
8
<
:
ð3:34Þ
Thus singular points of even k are stable, since both eigenvalues have negative
real parts. Singular points of odd k are unstable, since (1 + b
2 )
1/2 > b when
0 < b < 1, implying that one of the eigenvalues has a positive real part. These
results agree with the flow of orbits in Fig. 3.7.
3.4.5 On the Behavior of Orbits Near Singular Points
The above analysis of stability is based on Jacobian eigenvalues. These eigenvalues
merely reveal whether a singular point attracts or repels nearby orbits. To obtain a
more detailed picture on how orbits are disturbed in the vicinity of a singular point,
one needs to consider also the associated eigenvectors. The eigenvectors of the
linearized system will provide local information on orbits for the nonlinear system
as well, since near singular points the orbits of the nonlinear system are well
approximated by those of the linearized system.
Be careful to note that only the local behavior near singular points can be
assessed this way. To obtain a global orbital picture is kind of an art, requiring all
the local pictures to be tied together in the right manner; often this is not needed,
though.
Inspecting (3.27) it appears that the character of the linearized solutions depends
on whether the eigenvalues k are real or complex, and whether the eigenvalues are
distinct. Real eigenvalues cause the motion to grow or shrink exponentially in time,
whereas complex eigenvalues introduce oscillatory components in the motion
(since, with c, x 2 R: k = c + ix ) e
kt = e
c t (cos(xt) + i sin(xt))). Non-distinct
110
3 Nonlinear Vibrations: Classical Local Theory