Stability of singularly perturbed stochastic systems (Q1281009)

The author studies the rms stability of a linear singularly perturbed system of differential equations \[ dx=A_{11}(t)xdt+ A_{12}(t)ydt+ \bigl(B_{11} (t)x+B_{12}(t)y\bigr)dw \] \[ \varepsilon dy= A_{21} (t)xdt+ A_{22}(t)ydt +\sqrt \varepsilon\bigl(B_{21}(t)x+ B_{22}(t)y\bigr)dw \] whose coefficients are perturbed by a Gaussian white noise. The theory of integral manifolds of singularly perturbed systems is used to obtain the stability conditions. The effect of random forces on the stability of gyroscopic systems is discussed. The stability conditions are established. The possibility of replacing the complete equations of motion by the precession equations is considered, and a simple example shows that in the presence of the matrix of gyroscopic forces, the precession equations can provide an erroneous result.