Abstract
A Duffing oscillator excited by narrow-band random noise can exhibit multi-valued behaviour, with random jumps between two pseudo-stable states. In this paper an approximate form of the probability density function or the envelope of the response is obtained. State equations for the response envelope with the random excitation envelope as input are obtained by stochastic averaging. The joint probability density function of response envelope and input envelope can be found approximately for certain ranges of the system parameters. The response envelope probability density function shows two maxima in the cases where the response is multi-valued. Numerical simulation confirms the results.
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