Stability of a nonlinear discrete-time model of information dissemination under stochastic perturbations

Abstract

A nonlinear discrete-time model of information dissemination is considered and conditions for existence of a positive equilibrium of this system are obtained. It is shown that asymptotically stable positive equilibrium saves his stability under the influence of stochastic perturbations of the different types: small multiplicative perturbations, quickly fading multiplicative perturbations and quickly fading additive perturbations. Stability conditions are obtained using Lyapunov functions, are formulated in the terms of linear matrix inequalities (LMIs) and are illustrated by numerical simulations of solutions of the considered system. As an unsolved problem it is proposed to investigate the situation when stochastic perturbations fade on the infinity, but not very quickly. The proposed research method can be applied to investigate many other nonlinear mathematical models in various applications.