Stability Investigation of Systems of Nonlinear Stochastic Difference Equations

The aim of this research is to present a method for stability investigation of systems of nonlinear difference equations influenced by different types of stochastic perturbations. The proposed method is demonstrated on the stability investigation of a nonlinear discrete-time model of information dissemination. We obtain the conditions for this system's positive equilibrium. It is demonstrated that under the influence of stochastic perturbations of the various forms, including minor multiplicative perturbations, rapidly fading multiplicative perturbations, and rapidly fading additive perturbations, asymptotically stable positive equilibrium maintains its stability. Through the use of Lyapunov functions, linear matrix inequalities (LMIs), and numerical simulations of the system under consideration, stability requirements are determined. It is suggested to look at the scenario where stochastic disturbances fade on the infinity, but not very quickly, as an unresolved problem. The considered here method of stability investigation can be applied for many other nonlinear mathematical models described both by stochastic difference equations and by stochastic differential equations.