Non-stationary response of a van der Pol-Duffing oscillator under Gaussian white noise

This paper investigates the probability density evolution process of a van der Pol-Duffing oscillator under Gaussian white noise. A path integration method is employed with the Gauss–Legendre integration scheme. In the path integration method, the short-time Gaussian approximation scheme is used for computing the one-step transition probability density. Two cases are considered with slight nonlinearity or strong nonlinearity in displacement. The stationary and non-stationary responses of the oscillator are studied. Compared with the simulation result, the path integration method can present a satisfactory probability density function (PDF) solution for each case. Different probability density evolution processes are observed correspondingly. In the case of slight nonlinearity, the PDF undergoes a clockwise motion around the origin. The peak region gradually expands and the PDF eventually forms a circle. By contrast, the strong nonlinearity drives the oscillator to oscillate around the limit cycle. In such a case, the PDF rapidly forms a circle. The circle keeps its shape and develops until the oscillator becomes stationary. More complicated phenomena can be studied by the adopted path integration method.