Noise activated transitions among periodic states of a pendulum with a vertically oscillating pivot, mediated by a chaotic attractor

Abstract

It is well known that by harmonically displacing the point of suspension of a simple pendulum in a vertical direction, it is possible to produce a stable inverted state. Under certain conditions, this inverted state bifurcates into two distinct oscillations that mirror each other about the vertical. For some parameter values these two states, together with a third periodic oscillation that is symmetric about the downward direction, become embedded within a chaotic attractor. With added noise, the system dynamics can consist of endless patterns of escape from a given periodic attractor followed by capture by one of the three. After each escape and before the next capture, the system travels on the chaotic attractor. It is confirmed that the escape process is consistent with a picture of noise induced activation from the effective potential wells associated with each of the attractors.