It is supposed that the fractional difference equation x(n+1) = (mu + Sigma(k)(j=0)a(j)x(n-j))/(lambda + Sigma(k)(j=0)b(j)x(n-j)), n = 0, 1,..., has an equilibrium point (x) over cap and is exposed to additive stochastic perturbations type of sigma(x(n) - (x) over cap)xi(n+1) that are directly proportional to the deviation of the system state x(n) from the equilibrium point (x) over cap. 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.
Stability of Equilibrium Points of Fractional Difference Equations with Stochastic Perturbations
PATERNOSTER, Beatrice;
2008-01-01
Abstract
It is supposed that the fractional difference equation x(n+1) = (mu + Sigma(k)(j=0)a(j)x(n-j))/(lambda + Sigma(k)(j=0)b(j)x(n-j)), n = 0, 1,..., has an equilibrium point (x) over cap and is exposed to additive stochastic perturbations type of sigma(x(n) - (x) over cap)xi(n+1) that are directly proportional to the deviation of the system state x(n) from the equilibrium point (x) over cap. 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
