Suppression of perioddoubling and nonlinear parametric effects in periodically perturbed systems
Abstract
We consider the effect on a generic perioddoubling bifurcation of a periodic perturbation, whose frequency ω_{1} is near the perioddoubled frequency ω_{0}/2. The perturbation is shown to always suppress the bifurcation, shifting the bifurcation point and stabilizing the behavior at the original bifurcation point. We derive an equation characterizing the response of the system to the perturbation, analysis of which reveals many interesting features of the perturbed bifurcation, including (1) the scaling law relating the shift of the bifurcation point and the amplitude of the perturbation, (2) the characteristics of the system's response as a function of bifurcation parameter, (3) parametric amplification of the perturbation signal including nonlinear effects such as gain saturation and a discontinuity in the response at a critical perturbation amplitude, (4) the effect of the detuning (ω_{1}ω_{0}/2) on the bifurcation, and (5) the emergence of a closely spaced set of peaks in the response spectrum. An important application is the use of perioddoubling systems as smallsignal amplifiers, e.g., the superconducting Josephson parametric amplifier.
 Publication:

Physical Review A
 Pub Date:
 April 1986
 DOI:
 10.1103/PhysRevA.33.2525
 Bibcode:
 1986PhRvA..33.2525B
 Keywords:

 Branching (Mathematics);
 Dynamical Systems;
 Nonlinear Systems;
 Parametric Amplifiers;
 Period Doubling;
 Periodic Functions;
 Perturbation Theory;
 Systems Stability;
 Analog Simulation;
 Duffing Differential Equation;
 Josephson Junctions;
 Power Gain;
 Power Spectra;
 Scaling Laws;
 Physics (General);
 02.90.+p;
 03.20.+i;
 Other topics in mathematical methods in physics