Autoresonance

When driven with an appropriately chirped pump, the amplitude of certain nonlinear oscillators and waves can grow without bound. As the pump frequency is chirped towards the resonant frequency of the wave, the pump and the wave will phase lock at very low wave amplitude.  When the pump reaches the linear resonant frequency, significant amplitude growth begins.  Thereafter the amplitude of the wave will be such that the nonlinear wave frequency matches the pump frequency.  This phenomenon is called autoresonance, and is well known in accelerators, but has not been seen experimentally in a wave.  Experimental results are in remarkable agreement with theoretical predictions.

Autoresonance occurs in many systems.  Here we will briefly describe the theory for a pendulum, but most of our experiments were done with the diocotron wave in a non-neutral plasma.  The diocotron wave is a very high-Q oscillator which comes about because of the interaction of the plasma with the trap wall.  Click on the movie of a diocotron at right to get an expanded view.

How do you drive a nonlinear oscillator or wave to high amplitude?

If you drive the oscillator at its linear resonant frequency, it will grow for a while, but as its amplitude increases,  its frequency will shift and the drive will slip out of resonance.  The drive will then drift out of phase with the oscillator, and the oscillator's amplitude will decrease, eventually reaching zero.  The process will then repeat.

Driven Pendulum animation
You can use feedback to continuously adjust the driving frequency...this is how we push or pump a child's swing, subconsciously modifying the push or pump frequency to stay in phase with the swing.

But what if we can't (or don't want to) use feedback?  For example, we might not be able to measure the oscillator's phase, and could not adjust the frequency appropriately.

A very general property of driven, nonlinear oscillator systems is that, under certain conditions, they will automatically adjust their amplitude to stay matched with their drive.  By sweeping the drive frequency appropriately, we can control the oscillator's amplitude.  This phenomenon is called autoresonance.  An animation gives an example for a pendulum.

The theory behind autoresonance is given in the paper in the bibliography and in this poster.  For a given sweep rate, theory and experiments demonstrate that autoresonance only occurs when the drive amplitude is high enough. Surprisingly there is a very sharp drive amplitude threshold; below the threshold, the growth is very small.  Above the threshold the growth is unlimited.  Experiments measure the threshold to be very narrow, about 1% of the threshold drive amplitude.  The threshold drive amplitude V scales with the chirp rate a as V=a3/4.  This threshold and scaling law can be derived using action-angle variables.  The problem reduces to a Hamiltonian in which the oscillator looks like a pseudo-particle oscillating in a varying pseudo-potential.  So long as the pseudo-potential has wells, the particle will be trapped and autoresonance will occur.  But if the drive amplitude is too small, the pseudo-potential will flatten, the pseudo-particle will become untrapped, and autoresonance fails.  This process is illustrated  in this animation.  The drive of the red system on the left is above the threshold, and the pendulum is excited autoresonantly. The drive of the blue system on the right is below the threshold, and autoresonance does not occur.  The pseudo-potentials for both systems are shown on the bottom.

So far, we have discussed sweeping the drive frequency and keeping the rest of the system fixed.  Autoresonance can occur for any change in the system.  For instance, a pendulum can be driven autoresonant by shortening the pendulum length so that the pendulum's linear frequency passes through a fixed drive frequency.  This process is modeled in this animation.

Our experiments have been performed with the diocotron wave in non-neutral plasmas.  Typical results are shown in the graph below.