THE STABILITY OF STOCHASTICALLY PERTURBED ORBITAL MOTIONS

When investigating the orbital stability of non-linear stochastic syst ems, two forms of first-approximation systems (with noise of types I a nd II) are considered. The P-stability of first-approximation systems is defined. A necessary and sufficient condition for P-stability is th at the Lyapunov matrix differential equation should possess a periodic solution. An equivalent form is proposed for this criterion, using wh ich one can reduce the problem of stability for stochastic systems to determining the spectral radius of a certain positive operator. When t hat is done, lower (upper) bounds for the spectral radius yield necess ary (sufficient) conditions for stability. The possibilities of obtain ing constructive estimates are demonstrated for a system with one type II noise. A parametric stability criterion, which is a stochastic ana logue of the well-known Poincare criterion, is given for a two-dimensi onal system (the spectral radius is found in explicit form). Copyright (C) 1996 Elsevier Science Ltd.