Approximate solution of Mathieu-Duffing equation with bounded noise excitation

Mathieu-Duffing equation has been and is still used for parametric nonlinear vibrations in engineering. To our best knowledge, we have not found in the open literature any paper concerning the solution of Mathieu-Duffing equation with bounded noise excitation. We aim to obtain such solution. As is customary in vibration engineering, we use cos (Ωt + γW (t) + δ) as the bounded noise excitation term in Mathieu-Duffing equation, where W(t)) represents the Wiener process, γ represents the intensity of noise, and δ represents uniformly distributed stochastic variable in the interval (0,2π). In the full paper, we explain our solution in much detail; here we give only a briefing. We obtain an approximate theoretical solution with the method of multiple scales; this approximate theoretical solution is quite good as the non-dimensional coefficients in the Mathieu-Duffing equation are very small quantities. We also obtain solution with perturbation method, in which reasonable numerical values of the coefficients in Mathieu-Duffing equation need to be and can be easily assumed. The numerical solution agrees quite well with the approximate theoretical solution. Just how good is the agreement can be seen in Fig.2 of the full paper for the following values of the quantities in Mathieu-Duffing equation; h = 0.1, ω₀ = 1.0, γ = 0.5, β = 0.2, α = 0.1 (ε≪1, so coefficients εh,εβ, and εα are very small; ω₀² is the natural frequency, and γ is the intensity of noise). This agreement shows preliminarily that our approximate theoretical solution obtained with the method of multiple scales is reliable.