A hardening Duffing oscillator with

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

A hardening Duffing oscillator with ω0=1.000 rad s−1\omega_0=1.000\,\mathrm{rad\,s^{-1}}, β=0.0050 rad s−1\beta=0.0050\,\mathrm{rad\,s^{-1}} and ε=1.0 s−2 m−2\varepsilon=1.0\,\mathrm{s^{-2}\,m^{-2}} is driven at F0/m=5.0×10−3 m s−2F_0/m=5.0\times10^{-3}\,\mathrm{m\,s^{-2}} and swept to ω=1.050 rad s−1\omega=1.050\,\mathrm{rad\,s^{-1}}. Define G(A)≡A2[(ω02−ω2+34εA2)2+4β2ω2]−(F0/m)2,G(A)\equiv A^2\Big[(\omega_0^2-\omega^2+\tfrac34\varepsilon A^2)^2+4\beta^2\omega^2\Big]-(F_0/m)^2, so that G(A)=0G(A)=0 at each steady-state amplitude. Evaluate G(A)G(A) at A=0.020 mA=0.020\,\mathrm{m}, 0.050 m0.050\,\mathrm{m}, 0.300 m0.300\,\mathrm{m}, 0.365 m0.365\,\mathrm{m} and 0.450 m0.450\,\mathrm{m}, use the sign changes to say how many positive amplitudes solve G(A)=0G(A)=0 at this frequency, and state whether the response there is single- or triple-valued.

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