Near the onset of a period-doubling bifurcation, any dynamical system can be used to amplify perturbations near half the fundamental frequency: The closer the bifurcation point, the greater the amplification. An analytic expression for the frequency response curve is derived explicitly for the driven Duffing oscillator. Results of analog simulations are presented to check the main features of the theory. We propose that the superconducting Josephson parametric amplifier is an example of this amplification process. PACS numbers: 02.90.+p, 03.20. +i A great many systems, representing a wide variety of physical phenomena, are known to undergo perioddoubling instabilities. The goal of nonlinear dynamics is to determine what such systems have in common, irrespective of differences in the underlying physics.The most familiar results about period-doubling bifurcations concern the so-called period-doubling cascade, in which an infinite sequence of instabilities occur, culminating in chaotic dynamics. Most of the theoretical work has emphasized the dynamical behavior close to the onset of chaos, or within the chaotic regime.Researchers have found that several quantities obey scaling laws' 8 which are reminiscent (formally, at least) of scaling behavior observed in condensedmatter critical phenomena. On the experimental side, period-doubling sequences have proven to be fairly common, having been observed in electrical, 9 '4 optical, t5 hydrodynamic, '6 chemical, '6 and biological systems. '7'8 This Letter concerns the behavior of dynamical systems near the onset of a single period-doubling instability, far from a chaotic regime. This topic has received less attention then the period-doubling cascade, despite the fact that the mathematics of bifurcation theory tells us a great deal about the dynamics in such a situation. '9 Recently, work on the effect of random noise as a single instability is approached has shown that new broadband lines are induced in the power spectrum. 20 2' In effect, the input noise is greatly amplified, before the bifurcation, near frequencies where new sharp spectral lines appear after the bifurcation. Measurements on driven p-n junctions are in excellent agreement with the theoretical predictions. 2'The amplification of broadband noise at certain frequencies suggests that small coherent perturbations might also be amplified by systems near the onset of a period doubling. The purpose of this Letter is to show that this is indeed the case.Our basic result is as follows. Consider any dynamical system oscillating with period T, and suppose a parameter is adjusted so that the system is just before the onset of a period-doubling bifurcation. If one now couples in a small, monochromatic perturbation at frequency &o =Yr/T, the output of the system will have a large component at to. The magnitude of this amplification will grow substantially as the bifurcation point is approached.This result is independent of the specific dynamical system studied; however, to illustrate the ideas involved we focus on a spe...
