About Asymptotic Properties of Bilinear Difference Equations Under Stochastic Perturbations

During the last two decades, rational difference equations, in particular, rational bilinear difference equations, have become very popular in research. In this paper the asymptotic properties of one rational bilinear difference equation are rst studied under stochastic perturbations. It is assumed that stochastic perturbations are directly proportional to the deviation of the current value of the equation solution from one of its equilibria (the zero or nonzero). Some conditions are obtained by which the equilibrium under consideration is stable in probability or unstable. The obtained results are illustrated by numerical simulation of solutions of the considered stochastic difference equation. It is noted, that the research method used here can be applied to stability investigation of many other types of nonlinear difference equations with the order of nonlinearity higher than one. Some directions for further development of research are proposed. To readers attention some discussion about an unsolved problem of stabilization by noise for stochastic difference equations is also proposed. This problem, well known already for more than 50 years for stochastic differential equations, has not yet been solved until now for stochastic difference equations.