The upside-down pendulum theorem is
based on linear stability theory, so
only guarantees the stability of the inverted state with respect to
arbitrarily small disturbances. In fact, however, the inverted state can
recover from really rather large disturbances, provided the pendulums
are roughly aligned with one another initially. The first hint I got of
this was from a simple computer model, but Tom Mullin has subsequently
confirmed this kind of behaviour in the laboratory.

If, instead, the initial disturbance is such as to `buckle' the column of
pendulums, they typically collapse, though the computer model reveals
that if the buckling is not too severe they can, instead, end up
performing a kind of `dancing' oscillation about the upward
vertical.

We have not yet confirmed this peculiar feature in the actual
experiments, and it is one of a number of issues that still call for
further investigation.

*Proc. Roy. Soc. A Vol 448, pp 89-95, 1995*

(*See also Chapter 12 of *From Calculus to
Chaos)

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