Forced oscillations in nearly sinusoidal systems

Abstract
A large class of radio circuits which are analytically equivalent to an oscillatory network in parallel with a non-linear negative resistance, are represented fairly accurately by the differential equationv̈ − (α+βv−γv2)v + ω2v = Eω21 sinω1twhere, α/ω, β/ω, γ/ω are small. The behaviour of the solutions of this equation near resonance has been discussed by Appleton, van der Pol and others.The paper contains a more complete discussion of the synchronized and quasi-periodic solutions near resonance, their phases, amplitudes and energy, and also the way in which one type of stable solution gives way to another as the parameters of the system vary, for instance as the electromotive force or detuning vary. It is shown that the phase and amplitude favourable to synchronization are prolonged just before synchronization. This agrees with Appleton's experimental results. It is also found that hysteresis occurs. The decrease in energy with the decrease in detuning is explained by the fact that the phase favourable to synchronization is that which opposes the motion and is prolonged.

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