The Closed-Form Steady-State Probability Density Function of van der Pol Oscillator under Random Excitations

Abstract

The strongly nonlinear van der Pol oscillator represents a special challenge, which has prevented many methods from obtaining the closed-form solutions of the steady-state probability density functions (PDFs) in the literature. In this paper, we apply our recently developed method called the iterative method of weighted residue to analytically construct steady-state PDFs of the van der Pol oscillator under external Gaussian white noise excitation. The steady-state PDF is assumed to be an exponential function of polynomials in the state variables. The iterative method of weighted residue is used to compute the PDF. The iterative procedure that makes use of the obtained closed-form solutions of steady-state PDFs as the weighting function for the method improves the accuracy of the solution and the convergence of the solution process. The closed-form steady-state PDFs of strongly nonlinear van der Pol oscillator are presented in this paper, which were not available in the literature before, and are compared with those from the Monte Carlo simulations. The analytical and simulation results are in excellent agreement over a wide range of damping coefficients.