A FIRST APPROXIMATION OF THE QUASI-POTENTIAL IN PROBLEMS OF THE STABILITY OF SYSTEMS WITH RANDOM NONDEGENERATE PERTURBATIONS

The problem of a local description (near to a stationary point or orbi t) of the quasipotential-the Lyapunov function, used when analysing th e stability of a system with small non-degenerate random perturbations is considered. First approximations are constructed for quasipotentia ls in neighbourhoods of these invariant sets. The quadratic forms spec ifying these approximations are governed by certain matrices. The cons truction of these matrices is reduced to the solution of Lyapunov matr ix equations (which are algebraic in the case of stationary points, an d differential with periodic coefficients in the case of orbits).