A pendulum with vertically oscillating point of suspension is
an example of a
pendulum with periodic force
Linearizing around an equilibrium point
gives
the Mathieu type equation
where a(t) is a periodic function. If
and
,
this linear differential equation Lx=0 is called the
Mathieu equation. The differential operator L is called
the Mathieu operator.
For fixed ,
the set of parameters g for which x=0 is stable form
form a union of intervals called stable bands. The
complement is a set of unstable bands
.
The union of all unstable bands
is a union of Arnold tongues. If a(t) oscillates fast enough,
the point
can become stable. In the case of a piecewise
constant
for example, a pendulum of length 20cm
with suspension oscillation 1cm becomes stable at 43 Hertz
(see V. Arnold, Mathematical Methods of classical mechanics p. 122).
