Someone argues: "The amplitude equations for a pumped oscillator without damping give…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Nonlinear dynamics and chaos · Nonlinear dynamics and chaos

Modes, Packets and Media, Ch Driven and pumped nonlinear oscillators and the route to chaos

Problem

Someone argues: "The amplitude equations for a pumped oscillator without damping give a growth rate s=12(hω0/2)2−ν2s = \tfrac12\sqrt{(h\omega_0/2)^2 - \nu^2}. At exact tuning (ν=0\nu = 0) this is s=hω0/4s = h\omega_0/4, which is positive for any h>0h > 0. So the rest state is unstable for any nonzero modulation depth, and a pendulum with Q=200Q = 200 shaken at twice its natural frequency will start swinging however small the shaking amplitude is." Find the error in this argument, and give the correct threshold modulation depth hch_c for Q=200Q = 200.

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