On the investigation of mono-frequency oscillations in the system of quasi-linear differential-difference equations of second order with random perturbations (Q2768778)

The authors study mono-frequency oscillations described by the nonlinear differential-difference equations of second order with random perturbations NEWLINE\[NEWLINE\begin{multlined}\sum_{s=1}^{n}(a_{rs}\ddot x_{s}(t)+b_{rs}x_{s}(t)-c_{rs}x_{s}(t-\Delta))=\varepsilon f_{r}(x_1(t),\ldots,x_{n}(t),x_1(t-\tau),\ldots,x_{n}(t-\tau))\\ +\sqrt{\varepsilon}\sum_{s=1}^{n}g_{rs}(x_1(t),\ldots,x_{n}(t),x_1(t-\tau),\ldots,x_{n}(t-\tau)) \xi_{s}(t,\mu),\;r=1,2,\ldots,n. \end{multlined}NEWLINE\]NEWLINE Here, \(\varepsilon>0\) is a small parameter, \(\xi(t,\mu)=(\xi_{s}(t,\mu), s=1,2,\ldots,n)\) is a stationary random process converging to ``white noise'' as \(\mu\to 0\). NEWLINENEWLINENEWLINEThe authors construct a first approximation to a solution in the form \(x_{s}=\varphi_{s}^{(1)}a(t)\cos(\omega_1t+\theta(t))\), where \(\varphi_{s}\) is a normal function; the random amplitude \(a(t)\) and phase \(\theta(t)\) are defined from the system of equations \(da/dt=\varepsilon A_1(a,\xi(t,\mu),\varepsilon)\), \(d\theta/dt=\varepsilon B_1(a,\xi(t,\mu),\varepsilon)\). Equations for \(A_1, B_1\) are derived.