If damping is present (Fig. 3.7), the dissipation of energy causes orbits to spiral
towards the equilibriums at h = 0, 2p, …. With initial energies sufficiently large,
several full rotations may precede the spiralling.
The arrows in Fig. 3.6 and Fig. 3.7 indicate directions of motion for increasing
time t. This direction is easily established, by considering that h must increase in
the upper half-plane, where v > 0, and decrease in the lower half-plane, v < 0.
3.4.3 Singular Points
Certain points of a phase plane may correspond to states of static equilibrium. Such
points are termed singular points, fixed points, equilibriums or zeroes. All other
points of the phase plane are regular. For static equilibrium to occur we must have
ẋ = 0 in (3.16). Thus, singular points are located by solving the algebraic set of
equations f(x) = 0. We denote a singular point by ~ x, so that by definition f(~ x) = 0.
Fig. 3.6. Phase plane and orbits for the unforced pendulum: undamped case (b = 0)
Fig. 3.7. Phase plane and orbits for the unforced pendulum: damped case (0 < b < 1)
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3 Nonlinear Vibrations: Classical Local Theory