Qualitative analysis of conjugate harmonic oscillators under the random perturbations by ``white noise'' (Q2767625)

The author studies the behavior in the phase space of the image points of the stochastic differential equation NEWLINE\[NEWLINEd\xi_{i}(t)=a_{i}(\xi(t)) dt+ \sum_{k=1}^{2}b_{ik}(\xi(t)) dw_{k}(t), \quad t\geq 0,\;i=1,\ldots,4,NEWLINE\]NEWLINE where \(a_{i}(x), b_{ik}(x), i=1,\ldots,4, k=1,2\), are real non-random vector functions; \(w_{k}(t), k=1,2\), are independent one-dimensional Wiener processes. The following cases are considered: 1) \(b_{i2}(x)\equiv 0, i=1,\ldots,4\), the perturbation by ``white noise'' along the vector of phase velocity \((k_1x_2,-k_1 x_1,k_2 x_4,-k_2 x_3)\); 2) \(b_{12}(x)=b_{21}(x)=b_{32}(x)=b_{41}(x)=0\), the perturbation by two independent ``white noises'' along vectors of the phase velocities \((k_1x_2,-k_1x_1)\) and \((k_2x_4,-k_2x_3)\).