On difference equations with asymptotically stable 2-cycles perturbed by a decaying noise

Stabilisation of difference equations with noisy prediction-based control

“…We may ask the following of stochastically perturbed difference equations: (1) If the original (non-stochastic) map has chaotic or unknown dynamics, can we stabilise the equation by introducing a control with a stochastic component? (2) If the non-stochastic equation is either stable or has known dynamics (for example, a stable two-cycle [7]), do those dynamics persist when a stochastic perturbation is introduced? In this article, we consider both these questions in the context of prediction-based control (PBC, or predictive control).…”

“…For the modifications of the Beverton-Holt equation (6) x n+1 = Ax n 1 + Bx γ n , A > 1, B > 0, γ > 1, , x 0 > 0, n ∈ N 0 , and (7) x n+1 = Ax n (1 + Bx n ) γ , A > 1, B > 0, γ > 1, x 0 > 0, n ∈ N 0 Assumption 1.1 holds. Also, (6) and (7) satisfy Assumption 1.2 as long as the point at which the map on the right-hand side takes its maximum value is less than that of the point equilibrium. If Assumption 1.2 is not satisfied, the function is monotone increasing up to the unique positive point equilibrium, and thus all solutions converge to the positive equilibrium, and the convergence is monotone.…”

“…If all x n > K, we have a monotonically decreasing sequence. If we fix B in (6) and (7) and consider the growing A, the equation loses stability and experiences transition to chaos through a series of period-doubling bifurcations.…”

On convergence of solutions to difference equations with additive perturbations

“…• If there is a unique unstable positive equilibrium combined with a stable cycle, can the results of the present paper be extended to establish conditions and probabilities that the solution subject to stochastic perturbations will converge to this cycle? Two-cyclic behaviour of difference equations subject to eventually vanishing stochastic perturbations was studied in [7]. Define The Central Limit Theorem for (normalized and centralized) sum of independent random variables ζ 1 , ζ 2 , .…”

Stabilization of two cycles of difference equations with stochastic perturbations

“…However, our results would also hold for distributions with heavy tails, like polynomial, for example. More details about possible form of perturbation r and possible distributions involved can be found in [2], see also [3,10]. Papers [2,3,5] are focused on almost sure asymptotic stability of the equilibrium solution of stochastic difference equation with state independent perturbations.…”