A first approximation of the quasipotential in problems of the stability of systems with random nondegenerate perturbations (Q1359919)

A Lyapunov function of a special form -- the quasi-potential -- is used here to consider the stability of \[ dx= f(x)dt+ \varepsilon\sigma(x)dw(t), \] a system with small non-degenerate random perturbations. Specifically, the authors investigate the quasipotential in a neighbourhood of the stationary point and give an estimate of the error in finding the first approximation of the quasipotential. Consideration is also given to the stability problem when a closed curve (or orbit), rather than a point, is taken as the limiting variant set of the deterministic system \(dx= f(x)dt\). The existence and uniqueness of the solution to the Lyapunov matrix equation with periodic coefficients that is obtained are examined. The plane orbit (two-dimensional) case is also examined, and some illustrative examples are given.