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Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain


Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

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

Journal of Applied Nonlinear Dynamics 5(4) (2016) 495--502 | DOI:10.5890/JAND.2016.12.009

Lincong Chen; Jian-Qiao Sun

College of Civil Engineering, Huaqiao University, Xiamen, Fujian, 361021,China

School of Engineering, University of California at Merced, Merced, CA 95344, USA

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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.


This work is supported by the Natural Science Foundation of China through the Grants (11672111, 11172197, 11332008 and 11572215), by the National Science Foundation of Fujian Province under the Grant (2014J01014), Research Award Fund for Outstanding Young Researcher in Higher Education Institutions of Fujian Province and Research Fund for Excellent Young Scientific and technological Project of Huaqiao University under the Grant (ZQN-YX307). The first author would also like to thank the China Scholarship Council for sponsoring his studies at University of California, Merced through the grant (201408350008).


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