In this paper the results are reported of a set of numerical simulations applicable to the problem of particle acceleration by the beating of very high-frequency laser waves the ratio of driver frequency to plasma frequency close to 0 / p 30, including for the first time beat frequency chirping i.e., pump frequency linearly decreasing with time at various rates in order to counteract relativistic electron detuning of the plasma oscillation frequency and the resulting Rosenbluth–Liu amplitude limit. This amplitude limit can indeed be evaded, giving extremely high acceleration gradients of more than 25 GeV/m, for the case of numerical parameters relevant to the UCLA experiment University of California at Los Angeles. A chirp rate that is too fast leads to a sudden loss of frequency lock and a subsequent decay of the driven electric field, in agreement with the behavior of a similarly driven relativistic Lagrangian Rosenbluth–Liu oscillator. Even at chirp rates somewhat below this limiting value a remarkably localized and extraordinarily intense but as yet ill-understood electric field structure is formed, one that is unlikely to be compatible with the general requirement for smooth and coherent acceleration.