Introduction. In this paper, we are concerned with the Duffing oscillator, which has been applied to model many mechanical systems and has attracted much attention as a typical nonlinear system. When the system is under only a harmonic excitation or random one, two popular tools used to study such a nonlinear system are the averaging method and equivalent linearization method, respectively. The former was originally given by Krylov and Bogolyubov [1] and then it was developed by Bogolyubov and

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... itropolskiy [2] [3] [4] and was extended to systems under a random excitation with the works of Stratonovich [5], Khasminskii [6], and others, which were reviewed in survey paper by Mitropolskiy [3], Robert and Spanos [7] and Manohar [8]. The later, the stochastic equivalent linearization method, which has attracted many researchers due to its originality and capability for various applications in engineering, was first studies by Kazakov [9], who extended of deterministic problems to random problems. This method was also reviewed in some books by Roberts and Spanos [10], and Socha [11]. Recently, some approaches to the stochastic linearization have been proposed in Refs. [12][13][14]. In [13][14], for example, Anh have proposed a dual criterion of stochastic linearization method for single and multi-degree-offreedom nonlinear systems under white noise random excitations. The authors showed that the accuracy of the mean-square response is significantly improved when the nonlinearity increases. In a Duffing oscillator under periodic excitation, the phenomenon of subharmonic response has been known for years and has been described in many textbooks (see e.g. [15][16][17][18]) and works (see e.g. [19][20][21]). When the system is subjected to a combination of harmonic and random excitations, howeve r, to the au-(see e.g. [22][23][24][25]), there is no work on its subharmonic response. Thus, in this research, we present a technique to treat a one third order subharmonic response of a Duffing oscillator subjected to periodic and random excitations. The technique is a combination of the stochastic averaging method, the equivalent linearization method, and the technique of auxiliary function which yields the exact joint stationary probability den-