The third order superharmonic response of a Duffing oscillator to narrow band random excitation is analyzed. The analysis shows the effect of excitation bandwidth on the response and stability of the non-linear oscillator. In particular the analysis shows that multivalued superharmonic response can occur only when the excitation bandwidth is small. Stability of the response is examined by constructing a mean square phase plane from a time dependent Fokker-Planck equation and also by perturbing the stationary solutions.

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Journal Journal of Sound and Vibration
Davies, H.G. (H. G.), & Rajan, S. (1986). RANDOM SUPERHARMONIC RESPONSE OF A DUFFING OSCILLATOR. Journal of Sound and Vibration, 111(1), 61–70. doi:10.1016/S0022-460X(86)81423-7