Stability of a nonlinearly damped second-order system with randomly fluctuating restoring coefficient (Q1071931)

The author considers the second order ordinary differential equation \[ (*)\quad \ddot x+\epsilon g(\dot x)+\{1+\epsilon^{1/2}f(t)\}x=0. \] The damping force \(g(\dot x)\) is of the form \(g(\dot x)=\beta_ 0\dot x+\beta_{\alpha}\dot x| \dot x|^{\alpha}\) where \(\beta_{\alpha}>0\), \(\alpha\geq -1\). The parametric excitation f(t) is a mean zero stationary Gaussian process, and \(\epsilon\) is a small positive parameter. The initial conditions are \(x(0)=x_ 0\), \(\dot x(0)=\dot x_ 0\). For \(\epsilon \ll 1\), the solution of (*) may be approximated closely by a sinusoidal wave of slowly varying amplitude and phase. The amplitude a(t) satisfies a stochastic differential equation (as \(\epsilon >0)\). This equation is solved and various properties of a(t) are investigated.