Calculating the Lyapunov exponent for generalized linear systems with exponentially distributed elements of the transition matrix

A stochastic dynamic system of second order is considered. The system evolution is described by a dynamic equation with a stochastic transition matrix, which is linear in the idempotent algebra with operations of maximum and addition. It is assumed that some entries of the matrix are zero constants and all other entries are mutually independent and exponentially distributed. The problem considered is the computation of the Lyapunov exponent, which is defined as the average asymptotic rate of growth of the state vector of the system. The known results related to this problem are limited to systems whose matrices have zero off-diagonal entries. In the cases of matrices with a zero row, zero diagonal entries, or only one zero entry, the Lyapunov exponent is calculated using an approach which is based on constructing and analyzing a certain sequence of one-dimensional distribution functions. The value of the Lyapunov exponent is calculated as the average value of a random variable determined by the limiting distribution of this sequence.