Nonlinear inverted pendulum dynamics

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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