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.

We report on the autoresonant (nonlinear phase locking) manipulation of the diocotron mode in a non-neutral plasma. Autoresonance is a very general phenomena in driven nonlinear oscillator and wave systems, and allows us to control the amplitude of a nonlinear wave without the use of feedback. These are the first controlled laboratory studies of autoresonance in a collective plasma system. [S0031-9007(99)09167-X] PACS numbers: 52.25.Wz, 05.45.Xt, 52.35.Mw 4444 0031-9007͞99͞82(22)͞4444(4)$15.00

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Classical and quantum-mechanical phase locking transition in a nonlinear oscillator driven by a chirped frequency perturbation is discussed. Different limits are analyzed in terms of the dimensionless parameters P1 = ε/ √ 2m ω0α and P2 = (3 β)/(4m √ α) (ε, α, β and ω0 being the driving amplitude, the frequency chirp rate, the nonlinearity parameter and the linear frequency of the oscillator). It is shown that for P2 ≪ P1 + 1, the passage through the linear resonance for P1 above a threshold yields classical autoresonance (AR) in the system, even when starting in a quantum ground state. In contrast, for P2 ≫ P1 + 1, the transition involves quantum-mechanical energy ladder climbing (LC). The threshold for the phase-locking transition and its width in P1 in both AR and LC limits are calculated. The theoretical results are tested by solving the Schrodinger equation in the energy basis and illustrated via the Wigner function in phase space.

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