Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

Beatrice Paternoster,Leonid Shaikhet

doi:10.1155/2008/718408

Abstract

It is supposed that the fractional difference equation , has an equilibrium point and is exposed to additive stochastic perturbations type of that are directly proportional to the deviation of the system state from the equilibrium point . It is shown that 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 equilibrium points of the considered stochastic fractional difference equation. Numerous graphical illustrations of stability regions and trajectories of solutions are plotted.

Highlights

Equilibrium pointsRecently, there is a very large interest in studying the behavior of solutions of nonlinear difference equations, in particular, fractional difference equations 1–38
N 0, 1, . . . , has an equilibrium point x and is exposed to additive stochastic perturbations type of σ xn − x ξn 1 that are directly proportional to the deviation of the system state xn from the equilibrium point x
It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V

Summary

Recommended by Jianshe Yu

N 0, 1, . . . , has an equilibrium point x and is exposed to additive stochastic perturbations type of σ xn − x ξn 1 that are directly proportional to the deviation of the system state xn from the equilibrium point x. It is shown that known results in the theory of stability of stochastic difference equations that were obtained via V. Shaikhet general method of Lyapunov functionals construction can be successfully used for getting of sufficient conditions for stability in probability of equilibrium points of the considered stochastic fractional difference equation. There is a very large interest in studying the behavior of solutions of nonlinear difference equations, in particular, fractional difference equations 1–38. This interest really is so large that a necessity appears to get some generalized results. The stability of equilibrium points of the fractional difference equation μ xn 1 λ k j k j. Equation 1.1 generalizes a lot of different particular cases that are considered in 1–8, 16, 18–20, 22–24, 32, 35, 37

Put k k

And at last if

Aλ λA

2Bx Bx