Path integration of the Duffing–Rayleigh oscillator subject to harmonic and stochastic excitations W.X. Xie a, * , W. Xu a, * , L. Cai b a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China b College of Astronautics, Northwestern Polytechnical University, Xi’an, Shaanxi 710072, PR China Abstract This paper is focused on the Duffing–Rayleigh oscillator subject to harmonic and stochastic excitations using path integration based on the Gauss–Legendre integration scheme. First applying methods of harmonic balance and multiple scales to the deter- ministic case, the stabilities of the responses can be analyzed. Then the steady state peri- odic solution of probability density can be captured via path integration. At the same time, the changes of probability density induced by the intensities of harmonic and sto- chastic excitations are discussed in three cases. Ó 2005 Elsevier Inc. All rights reserved. Keywords: Path integration; Duffing–Rayleigh oscillator; Method of harmonic balance; Method of multiple scales; Harmonic and stochastic excitation 0096-3003/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2005.01.095 * Corresponding authors. E-mail addresses: xiewy8899@eyou.com (W.X. Xie), weixu@nwpu.edu.cn (W. Xu). Applied Mathematics and Computation 171 (2005) 870–884 www.elsevier.com/locate/amc