The stability of stochastically perturbed orbital motions

Abstract

When investigating the orbital stability of non-linear stochastic systems, two forms of first-approximation systems (with noise of types I and II) are considered. The P-stability of first-approximation systems is defined. A necessary and sufficient condition for P-stability is that the Lyapunov matrix differential equation should possess a periodic solution. An equivalent form is proposed for this criterion, using which one can reduce the problem of stability for stochastic systems to determining the spectral radius of a certain positive operator. When that is done, lower (upper) bounds for the spectral radius yield necessary (sufficient) conditions for stability. The possibilities of obtaining constructive estimates are demonstrated for a system with one type II noise. A parametric stability criterion, which is a stochastic analogue of the well-known Poincaré criterion, is given for a two-dimensional system (the spectral radius is found in explicit form).