Stability of Equilibria of Fractional Difference Equations under Stochastic Perturbations

Abstract

It is supposed that the fractional difference equation$$x_{n+1}=\frac{\mu+\sum\limits^k_{j=0}a_jx_{n-j}}{\lambda+\sum\limits^k_{j=0}b_jx_{n-j}}, \qquad n=0,1,...,$$has an equilibrium \(\hat x\) and is exposed to additive stochastic perturbations that are directly proportional to the deviation of the system state \(x_n\) from the equilibrium \(\hat x\), i.e., are of the type of \((x_n-\hat x)\xi_{n+1}\). It is shown how the known results in the theory of stability of stochastic difference equations that were obtained via V. Kolmanovskii and L. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibria of the considered stochastic fractional difference equation. To demonstrate the obtained results numerical simulations of equations solutions are made and numerous graphical illustrations of stability regions and trajectories of solutions are plotted. The proposed method can be successfully used for stability investigation of other type of nonlinear difference equations both with discrete and continuous time. And this is of particular interest, since in the last decade, studies of difference equations, in particular, rational difference equations, have been continued and actively developed.