Autoresonant (nonstationary) excitation of pendulums, Plutinos, plasmas, and other nonlinear oscillators

Abstract

A weakly driven pendulum cannot be strongly excited by a fixed frequency drive. The only way to strongly excite the pendulum is to use a drive whose frequency decreases with time. Feedback is often used to control the rate at which the frequency decreases. Feedback need not be employed, however; the drive frequency can simply be swept downwards. With this method, the drive strength must exceed a threshold proportional to the sweep rate raised to the 3/4 power. This threshold has been discovered only recently, and holds for a very broad class of driven nonlinear oscillators. The threshold may explain the abundance of 3:2 resonances and dearth of 2:1 resonances observed between the orbital periods of Neptune and the Plutinos (Pluto and many of the Kuiper Belt objects), and has been extensively investigated in the Diocotron system in pure-electron plasmas.

Autoresonant control of the magnetization switching in single-domain nanoparticles

“…This approach was originally devised for a simple nonlinear oscillator (e.g., a pendulum) driven by a chirped force with a slowly varying frequency [15][16][17]. If the driving amplitude exceeds a certain threshold, then the nonlinear frequency of the oscillator stays locked to the excitation frequency, so that the resonant match is never lost (until, of course, some other effects start to kick in).…”

Manipulation of Autoresonant Intrinsic Localized Modes in MEMs Arrays

“…The end result is that the oscillation frequency is slightly higher than the top of the band, and this resulting ILM is continuously frequency locked to the driver [14]. In this autoresonant state [15] the amplitude of the ILM is controlled by the frequency of the driver. At the heart of the experimental measurement method is an optical lever.…”

“…1. The PZT is turned on at t = 0 ms and then its frequency is chirped up from the highest linear extended mode frequency (top of the band) to above it to reach the high amplitude autoresonant ILM state [15]. In the middle time interval of this figure there are many highly excited regions but after about 12 ms three stationary ILMs remain.…”

Experimental Study of Parametric Autoresonance in Faraday Waves

“…The precise form of the chirp is unimportant once its sign is correct, and the chirp rate is below a critical value. First predicted for harmonic forcing, autoresonance has found many applications [2]. The technique was extended to weakly dissipative oscillators [3] and to nonlinear waves and vortices in non-dissipative systems [4].…”

“…For small m andβ, the PAR wave growth corresponds to the stable quasi-fixed point of Eqs. (2). To leading order…”