Averaging in randomly perturbed multifrequency nonlinear systems (Q1359917)

The author proves some kind of averaging principle for the dynamics system \[ \dot x=\varepsilon F(x,\theta)\xi(t)+ \varepsilon^2G(x,\theta),\;\dot\theta= \omega(x)+ \varepsilon H(x,\theta)\xi(t)+ \varepsilon^2D(x,\theta), \] where \(\varepsilon\) is a small parameter; \(x\in\mathbb{R}^n\); \(\theta\in\mathbb{R}^m\), \(\xi(t)\in \mathbb{R}^l\) is a stationary stochastic process with zero mean satisfying the mixing condition, and \(F\), \(H\), \(G\), \(D\) are matrices of the appropriate dimensions which are \(2\pi\)-periodic functions of each of the components of the vector \(\theta\) and sufficiently smooth. An implicit assumption in this problem is that all the frequencies \(\omega_k\) are of the same order of magnitude. In the case when \(|\omega_r(x)/\omega_l(x)|= O(\varepsilon)\), the equations of motion depend on both slowly and rapidly rotating phases. This general case may be written as \[ \dot x=\varepsilon F(x,\varphi,\theta)\xi(t),\;\dot\varphi= \varepsilon\Omega(x)+ \varepsilon\Phi(x, \varphi,\theta)\xi(t),\;\dot\theta= \omega(x)+\varepsilon H(x\varphi,\theta)\xi(t), \] and the averaging principle for such systems is proved. The non-resonant and the resonant cases are considered. An example, namely, the perturbed motion of a gyroscope in gimbals is presented.