Fig. 3.8(e)–(f) shows the corresponding orbits in a (Re(u 1 ), Im(u 1 )) plane (those
for u 2 are similar). If Re(k) = 0, i.e. the eigenvalues are purely imaginary, the orbits
are circles of radius r 0 , and the singular point is called a center (Fig. 3.8(e)).
A center may be considered stable or unstable, according to various definitions
of stability. For the moment we bypass this discussion by calling a center marginally or neutrally stable. Centers and other singular points with Re(k) = 0, are
significant for the study of bifurcations – the qualitative shifts in system behavior
that we shall consider further in Chap. 5.
If Re(k) 6 ¼ 0 the orbits spiral around the singular point, which is then called a
focus (Fig. 3.8(f)). For Re(k) < 0 the orbits approach the singular point, and the
focus is stable. This completes our discussion of possible types of singular points
for two-dimensional systems with diagonal Jordan form.
Orbits for Non-diagonal Jordan Forms The non-diagonal Jordan form (3.43)
applies when |[a 1 a 2 ]| = 0, that is, when the eigenvalues of J(~ x) are real and equal
and the eigenvectors are linearly dependent. The system (3.37) becomes:
_
u 1 ¼ ku 1 þ u 2 ; _
u 2 ¼ ku 2 ;
ð3:50Þ
with solution
u 1 ðtÞ ¼ u 10 þ u 20 t
ð
Þ e
kt
; u 2 ðtÞ ¼ u 20 e
kt
;
ð3:51Þ
or, by eliminating t:
u 1 ¼
u 10
u 20
þ
1
k
ln
u 2
u 20
u 2
ð3:52Þ
The orbits (u 1 (t),u 2 (t) are shown in Fig. 3.8(d). It appears from (3.44) that two
half-orbits (u 2 (t) = 0, t 6 ¼ 0) coincide with the u 1 -axis, whereas there are no orbits
along the u 2 -axis. The singular point is a node, being stable for k < 0.
Orbital Topology for the Pendulum Case We now return then to the singular
points (h,v) = (kp,0) of the unforced pendulum.
In the undamped case, according to Eq. (3.33), the two eigenvalues are purely
imaginary for even k, and real and distinct with different signs for odd k. Hence,
according to Fig. 3.8, the singular points corresponding to even k are centers,
whereas those corresponding to odd k are saddles. Indeed, this knowledge is sufficient for sketching the phase plane orbits shown in Fig. 3.6. First we would draw
the local arrangements of orbits near the individual centers and saddles. Next,
observing that orbits must be smooth and cannot intersect, we could connect the
stable and unstable saddle branches into heteroclinic orbits, and finally sketch the
outermost running orbits, running outside the heteroclinic loops.
In the damped case, according to Eq. (3.34), the eigenvalues are complex with
negative real parts for even k, and real and distinct with different signs for odd
k. Hence, according to Fig. 3.8, singular points corresponding to even k are stable
3.4 Qualitative Analysis of the Unforced Response
115
for u 2 are similar). If Re(k) = 0, i.e. the eigenvalues are purely imaginary, the orbits
are circles of radius r 0 , and the singular point is called a center (Fig. 3.8(e)).
A center may be considered stable or unstable, according to various definitions
of stability. For the moment we bypass this discussion by calling a center marginally or neutrally stable. Centers and other singular points with Re(k) = 0, are
significant for the study of bifurcations – the qualitative shifts in system behavior
that we shall consider further in Chap. 5.
If Re(k) 6 ¼ 0 the orbits spiral around the singular point, which is then called a
focus (Fig. 3.8(f)). For Re(k) < 0 the orbits approach the singular point, and the
focus is stable. This completes our discussion of possible types of singular points
for two-dimensional systems with diagonal Jordan form.
Orbits for Non-diagonal Jordan Forms The non-diagonal Jordan form (3.43)
applies when |[a 1 a 2 ]| = 0, that is, when the eigenvalues of J(~ x) are real and equal
and the eigenvectors are linearly dependent. The system (3.37) becomes:
_
u 1 ¼ ku 1 þ u 2 ; _
u 2 ¼ ku 2 ;
ð3:50Þ
with solution
u 1 ðtÞ ¼ u 10 þ u 20 t
ð
Þ e
kt
; u 2 ðtÞ ¼ u 20 e
kt
;
ð3:51Þ
or, by eliminating t:
u 1 ¼
u 10
u 20
þ
1
k
ln
u 2
u 20
u 2
ð3:52Þ
The orbits (u 1 (t),u 2 (t) are shown in Fig. 3.8(d). It appears from (3.44) that two
half-orbits (u 2 (t) = 0, t 6 ¼ 0) coincide with the u 1 -axis, whereas there are no orbits
along the u 2 -axis. The singular point is a node, being stable for k < 0.
Orbital Topology for the Pendulum Case We now return then to the singular
points (h,v) = (kp,0) of the unforced pendulum.
In the undamped case, according to Eq. (3.33), the two eigenvalues are purely
imaginary for even k, and real and distinct with different signs for odd k. Hence,
according to Fig. 3.8, the singular points corresponding to even k are centers,
whereas those corresponding to odd k are saddles. Indeed, this knowledge is sufficient for sketching the phase plane orbits shown in Fig. 3.6. First we would draw
the local arrangements of orbits near the individual centers and saddles. Next,
observing that orbits must be smooth and cannot intersect, we could connect the
stable and unstable saddle branches into heteroclinic orbits, and finally sketch the
outermost running orbits, running outside the heteroclinic loops.
In the damped case, according to Eq. (3.34), the eigenvalues are complex with
negative real parts for even k, and real and distinct with different signs for odd
k. Hence, according to Fig. 3.8, singular points corresponding to even k are stable
3.4 Qualitative Analysis of the Unforced Response
115