The singular points of the pendulum equation (3.14) are obtained by solving the
system _
h ¼ _
v ¼ 0. This gives v = 0 and sinh = 0, that is:
ð ~ h; ~ vÞ ¼ ðkp; 0Þ; k ¼ . . . À 1; 0; 1; . . .
ð3:22Þ
In Fig. 3.6 and Fig. 3.7 the singular points are marked by filled or unfilled
circles. Note that in the undamped case orbits encircle the singular points at h = 0,
2p, …, whereas in the damped case these points attract nearby orbits.
For deterministic systems only a single orbit x(t) can pass through each point of
state space. Otherwise orbits could intersect themselves, or other orbits, and the
state of the system would not be uniquely determined by the initial conditions. This
property also holds for singular points: only a single orbit may pass through.
Reconsidering Fig. 3.6 and Fig. 3.7, it may seem as if several orbits pass through
the singular points of the pendulum system. However, the only orbit passing
through a singular point is the singular orbit. You may imagine the singular orbit as
a straight line extending through a singular point and behind the paper for t !
+ ∞, and out of the paper for t ! –∞. Other orbits may at most approach this
orbit, that is, it will take them infinitely long to actually reach it. Non-singular orbits
are called regular.
One should be careful here to distinguish between the notions of state space and
phase plane. The above observations on intersections of orbits apply to orbits in
thestate space, which is spanned by the state-variables of a set of autonomous,
first-order differential equations. Phase planes are spanned by any two
state-variables. For the unforced pendulum, since there are only two autonomous
equations, the state space happens to be also a phase plane. Thus, for
two-dimensional systems, orbits do not intersect in the phase plane. Now, considering instead the forced pendulum, we would have to write three autonomous
equations
2 . Then orbits in the three-dimensional state space would not intersect,
while orbits projected on a phase plane probably would; they are only projections
of the state space orbits to the plane.
Orbits that connect distinct singular points are called heteroclinic. In Fig. 3.6 the
orbits connecting the singular points at h = –p, p, are heteroclinic. Other orbits
(though none for the pendulum) are homoclinic; they loop from a singular point and
back to that same point.
3.4.4 Stability of Singular Points
A stable singular point attracts nearby orbits, an unstable singular point repels
orbits, whereas a marginally/neutrally stable singular point acts as a center, neither
repelling nor attracting orbits. Certain singular points have a mixed kind of stability,
2 _
h ¼ v; _
v ¼ 2bx 0 v À x
2
0 sin h þ qX
2 cos z, and _
z ¼ X.
3.4 Qualitative Analysis of the Unforced Response
107
system _
h ¼ _
v ¼ 0. This gives v = 0 and sinh = 0, that is:
ð ~ h; ~ vÞ ¼ ðkp; 0Þ; k ¼ . . . À 1; 0; 1; . . .
ð3:22Þ
In Fig. 3.6 and Fig. 3.7 the singular points are marked by filled or unfilled
circles. Note that in the undamped case orbits encircle the singular points at h = 0,
2p, …, whereas in the damped case these points attract nearby orbits.
For deterministic systems only a single orbit x(t) can pass through each point of
state space. Otherwise orbits could intersect themselves, or other orbits, and the
state of the system would not be uniquely determined by the initial conditions. This
property also holds for singular points: only a single orbit may pass through.
Reconsidering Fig. 3.6 and Fig. 3.7, it may seem as if several orbits pass through
the singular points of the pendulum system. However, the only orbit passing
through a singular point is the singular orbit. You may imagine the singular orbit as
a straight line extending through a singular point and behind the paper for t !
+ ∞, and out of the paper for t ! –∞. Other orbits may at most approach this
orbit, that is, it will take them infinitely long to actually reach it. Non-singular orbits
are called regular.
One should be careful here to distinguish between the notions of state space and
phase plane. The above observations on intersections of orbits apply to orbits in
thestate space, which is spanned by the state-variables of a set of autonomous,
first-order differential equations. Phase planes are spanned by any two
state-variables. For the unforced pendulum, since there are only two autonomous
equations, the state space happens to be also a phase plane. Thus, for
two-dimensional systems, orbits do not intersect in the phase plane. Now, considering instead the forced pendulum, we would have to write three autonomous
equations
2 . Then orbits in the three-dimensional state space would not intersect,
while orbits projected on a phase plane probably would; they are only projections
of the state space orbits to the plane.
Orbits that connect distinct singular points are called heteroclinic. In Fig. 3.6 the
orbits connecting the singular points at h = –p, p, are heteroclinic. Other orbits
(though none for the pendulum) are homoclinic; they loop from a singular point and
back to that same point.
3.4.4 Stability of Singular Points
A stable singular point attracts nearby orbits, an unstable singular point repels
orbits, whereas a marginally/neutrally stable singular point acts as a center, neither
repelling nor attracting orbits. Certain singular points have a mixed kind of stability,
2 _
h ¼ v; _
v ¼ 2bx 0 v À x
2
0 sin h þ qX
2 cos z, and _
z ¼ X.
3.4 Qualitative Analysis of the Unforced Response
107