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Stability of equilibrium points of fractional difference equations with stochastic perturbations. (English) Zbl 1149.39007

Summary: It is supposed that the fractional difference equation \(x_{n+1}=(\mu +\sum _{j=0}^{k} a_{j}x_{n - j})/(\lambda +\sum _{j=0}^{k}b_{j}x_{n - j}), n=0,1,\dots ,\) has an equilibrium point \(\hat x\) and is exposed to additive stochastic perturbations type of \(\sigma (x_{n} - \hat x)\xi _{n+1}\) that are directly proportional to the deviation of the system state \(x_{n}\) from the equilibrium point \(\hat x \). It is shown that known results in the theory of stability of stochastic difference equations that are obtained via V. Kolmanovskii and L. Shaikhet’s general method of Lyapunov functionals construction [Dynamical systems and applications. World Sci. Ser. Appl. Anal. 4, 397–439 (1995; Zbl 0846.93083); Math. Comput. Modelling 36, No. 6, 691–716 (2002; Zbl 1029.93057)] 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.

MSC:

39A11 Stability of difference equations (MSC2000)
39A20 Multiplicative and other generalized difference equations
93E15 Stochastic stability in control theory
37H10 Generation, random and stochastic difference and differential equations
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