Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations

• Main Authors: Shaikhet Leonid, Paternoster Beatrice
• Format: Article
• Language: English
• Published: SpringerOpen 2008-01-01
• Series: Advances in Difference Equations
• Online Access: http://www.advancesindifferenceequations.com/content/2008/718408

<p/> <p>It is supposed that the fractional difference equation <inline-formula> <graphic file="1687-1847-2008-718408-i1.gif"/></inline-formula>, <inline-formula> <graphic file="1687-1847-2008-718408-i2.gif"/></inline-formula> has an equilibrium point <inline-formula> <graphic file="1687-1847-2008-718408-i3.gif"/></inline-formula> and is exposed to additive stochastic perturbations type of <inline-formula> <graphic file="1687-1847-2008-718408-i4.gif"/></inline-formula> that are directly proportional to the deviation of the system state <inline-formula> <graphic file="1687-1847-2008-718408-i5.gif"/></inline-formula> from the equilibrium point <inline-formula> <graphic file="1687-1847-2008-718408-i6.gif"/></inline-formula>. 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.</p>