Arnold pendulum

Inverted pendulum

A pendulum with vertically oscillating point of suspension is an example of a pendulum with periodic force

$\begin{displaymath}\ddot{x} = - g \; \sin(x) + \epsilon F(t,x) \; . \end{displaymath}$

Linearizing around an equilibrium point $x=\pi$ gives the Mathieu type equation

$\begin{displaymath}\ddot{x} = -g (1+ \epsilon a(t)) x \end{displaymath}$

where a(t) is a periodic function. If $g\leq 0$ and $a(t)=\cos(t)$ , this linear differential equation Lx=0 is called the Mathieu equation. The differential operator L is called the Mathieu operator.

For fixed $\epsilon$ , 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 $B_{\epsilon}$ . The union of all unstable bands $\bigcup_{\epsilon} B_{\epsilon}$ is a union of Arnold tongues. If a(t) oscillates fast enough, the point $x=\pi$ can become stable. In the case of a piecewise constant $a(t)= \pm d$ 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).

Animation