Stochastically perturbed Hopf bifurcation (Q1097391)

A two-dimensional system of differential equations \[ \frac{dy}{dt}=F(y,\sigma)+\epsilon^{1/2}[A_ 1(\sigma)f(t)+A_ 2(\sigma)g(t)]y \] is considered where F is a vector function analytic in y and \(\sigma\), \(\sigma\) is a real-valued parameter, \(A_ 1(\sigma)\), \(A_ 2(\sigma)\) are \(2\times 2\)-matrices which depend only on \(\sigma\), f(t), g(t) are uncorrelated stationary processes with zero mean values, \(\epsilon\) is a small parameter and \(\epsilon \ll 1\). The processes f(t), g(t) have small correlation time and smooth spectral density. The system undergoes a Hopf bifurcation at \(\sigma =\sigma_ c\). The paper studies almost sure and moment stability of the trivial and bifurcating solutions.