Keywords: Fokker-Planck equation, stochastic averaging, Galerkin approximation, numerical solution, van der Pol type oscillator, partial amplitudes.

The response of slender engineered structures in close proximity to the lock-in frequency region exhibits multiple dominant frequencies that contribute to the quasi-periodic nature of the response. The difference in individual dominant frequencies increases significantly with increasing distance from the lock-in region. This effect alters the character of the response from apparently non-stationary to quasi-periodic, with the frequency of beating varying as the distance from the locking interval changes. In the presence of combined random and harmonic excitation, the response character varies between stationary, cyclo-stationary, and non-stationary, depending on the intensity of the stochastic component. While the probabilistic characteristics of a non-linear Single Degree of Freedom oscillator of the van der Pol type system on a slow time scale can be described using partial amplitudes of the response, this paper specifically focuses on the non-stationary case. The solution to the Fokker-Planck equation for the cross-Probability Density Function of the partial amplitudes is determined using the Galerkin approximation. For this purpose, orthogonal polynomial basis functions %and the exponential-polynomial-closure method are utilized and assessed.

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