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).