Classical Mechanics (Goldstein) · Problem 8

Classical Chaos — Herbert Goldstein, Charles Poole, John Safko

Physics — Mechanics · duffing-oscillator

Problem

The general forced damped oscillator equation studied by F. Duffing in 1918 (Duffing oscillator) can be written as d2xdt2+2γdxdt+αx+βx3=Fcos⁡ωt.\frac{d^2x}{dt^2} + 2\gamma \frac{dx}{dt} + \alpha x + \beta x^3 = F\cos\omega t. (a) Take α=1\alpha = 1, β=0.2\beta = 0.2, γ=0\gamma = 0, F=4.0F = 4.0, [dx/dt]t=0=0[dx/dt]_{t=0} = 0 and choose a set of values of xt=0x_{t=0} and ω\omega to show that the amplitude (absolute magnitude of the maximum xx) of the steady-state oscillation shows hysteresis. This is best done by plotting the behavior for increasing ω\omega until there is a jump in the amplitude and then continuing the plot for slowly decreasing ω\omega from a value slightly larger than where the jump occurred. (b) Having solved part (a), pick a value of ω\omega in the range of the jump and slightly vary FF to determine how the amplitude varies for a fixed ω\omega as FF is changed.

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