Calculation of the asymptotic characteristics of a stochastic synchronized dynamic system

Abstract

This paper is concerned with the model of a stochastic dynamic system with synchronized events. The dynamics of the system is described by a generalized linear equation with a matrix involving one random entry on the diagonal; the remaining entries are nonnegative constants that are related by some conditions. The problem is to determine the mean asymptotic growth rate of the state vector (the Lyapunov exponent) of the system. The solution depends upon a change of variables, as a result of which new random variables are introduced instead of the random coordinates of the state vector. It is shown that in many cases by an appropriate choice of new variables one may reduce the problem to examining only one sequence of random variables given by a recurrence equation of a certain form, which depends only on two of three constants in the matrix of the system. The construction of such a system of random variables is followed by examination of its convergence. The Lyapunov exponent of a system is obtained as the mean value of the limit distribution of a sequence.