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Multiple-nodding oscillations of a driven inverted pendulum

It has long been known that a simple rigid pendulum can be stabilized against small disturbances in its upside-down state by rapid vertical oscillations of its pivot. Over part of the parameter range for which this happens we find that there is a second, quite different way in which the pendulum can avoid falling over, namely by getting trapped in a limit cycle oscillation about the upward vertical. Each upside-down oscillation of this kind has a distinctive 'multiplenodding' character.