Large-Signal Analysis of a Parametric Second-Subharmonic Oscillator

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Abstract
This theoretical analysis finds the steady-state and transient second-subharmonic oscillations of a parametric oscillator described by\ddot{q} + 2k\dot{q} + (1 - \delta) q + bq^2 = (3 + \delta) \cos 2z. Harmonic balance yields expressions, with experimentally observed double-valued properties, for the steady-state subharmonic amplitude in terms of pump amplitude and frequency. Reducing the pump frequency below twice the forced resonant frequency (i.e.\delta< 0) provides larger subharmonic amplitude at the expense of a larger minimum pump amplitude required for oscillation. The transient, analyzed by harmonic balance assuming a slowly varying subharmonic amplitude, is portrayed by trajectories in the in-phase vs quadrature amplitude plane. Those singular points which are closer to this plane's origin than the steady-state points are classified using an index theorem. The following results explain the "hysteresis" effects in the double-valued steady state: The origin is a saddle point (unstable) when it is the only such singularity; it is either a stable node or focus when two other such singularities appear, these being saddle points. Numerical computations of trajectories for parameters giving a saddle-point origin show that although a measure of the 10-90 per cent response time is given by\tauIn 9, (\taubeing the time constant at the origin, measured in subharmonic cycles), this response time does not increase proportionately to\tauas the pump frequency is decreased. The ratiob/2k = 2provides unexcessive overshoot in attaining steady state without severely degrading\tau. Experimental checks of the transien and steady state are presented.

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