Stability of equilibrium points of fractional difference equations with stochastic perturbations.

*(English)*Zbl 1149.39007Summary: 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, e.g., of Lyness type |

93E15 | Stochastic stability in control theory |

37H10 | Generation, random and stochastic difference and differential equations |

##### Keywords:

fractional difference equation; equilibrium point; additive stochastic perturbations; stability; stochastic difference equations; Lyapunov functionals##### References:

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