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The stochastic averaged equations for energy envelope and amplitude envelope of single-degree-of-freedom (SDOF) strongly nonlinear oscillators subject to Poisson white noise excitations are derived.The difference between these two kind of stochastic averaged equations is discussed.The averaged generalized Fokker-Planck-Kolmogorov (GFPK) equations associated with these two kinds of stochastic averaged equations are derived and a perturbation procedure is proposed for obtaining the approximate stationary solution of the averaged GFPK equations.As an example, the approximate stationary probability density function for the response of Duffing-van der Pol oscillator to both external and parametric excitations of Poisson white noise are obtained by using these two stochastic averaging procedures and confirmed by Monte Carlo simulations.