On stochastic stabilization of difference equations

@article{Appleby2006OnSS,
title={On stochastic stabilization of difference equations},
author={John A. D. Appleby and Xuerong Mao and Alexandra Rodkina},
journal={Discrete and Continuous Dynamical Systems},
year={2006},
volume={15},
pages={843-857}
}
• Published 1 April 2006
• Mathematics
• Discrete and Continuous Dynamical Systems
We consider unstable scalar deterministic difference equation $x_{n+1}=x_n(1+a_nf(x_n))$, $n\ge 1$, $x_0=a$. We show how this equation can be stabilized by adding the random noise term $\sigma_ng(x_n)\xi_{n+1}$ where $\xi_n$ takes the values +1 or -1 each with probability $1/2$. We also prove a theorem on the almost sure asymptotic stability of the solution of a scalar nonlinear stochastic difference equation with bounded coefficients, and show the connection between the noise…
Stabilization of difference equations with noisy proportional feedback control
• Mathematics
• 2016
Given a deterministic difference equation $x_{n+1}= f(x_n)$, we would like to stabilize any point $x^{\ast}\in (0, f(b))$, where $b$ is a unique maximum point of $f$, by introducing proportional
Stochastic difference equations with the Allee effect
• Mathematics
• 2016
For a truncated stochastically perturbed equation $x_{n+1}=\max\{ f(x_n)+l\chi_{n+1}, 0 \}$ with $f(x) f(H)>m$. As the amplitude grows, an interval $(m-\varepsilon, m+\delta)$ of initial values
On Global Asymptotic Stability of Nonlinear Stochastic Difference Equations with Delays
• Mathematics
• 2006
We consider the stochastic difference equation Xn+1 = aF ( Xn − k ∑ l=1 blXn−l ) + g(n,Xn, Xn−1, . . . , Xn−k)ξn+1, n ∈ N0 (0.1) with arbitrary initial conditionsX0, X−1, . . . , X−k ∈ R, non-linear
On almost sure asymptotic periodicity for scalar stochastic difference equations
• Mathematics
• 2017
We consider a perturbed linear stochastic difference equation 1X(n+1)=a(n)X(n)+g(n)+σ(n)ξ(n+1),n=0,1,…,X0∈R, X(n+1)=a(n)X(n)+g(n)+\sigma(n)\xi(n+1), \quad n=0, 1, \dots, \qquad X_{0}\in\mathbb{R},
On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations
• Mathematics
• 2008
We consider the nonlinear stochastic difference equation Here, (ξ n ) n ∈ ℕ is a sequence of independent random variables with zero mean and unit variance and with distribution functions F n . The
Stabilization and Destabilization of Nonlinear Differential Equations by Noise
• Mathematics
IEEE Transactions on Automatic Control
• 2008
It is shown that when f is locally Lipschitz, a function g can always be found so that the noise perturbation g(X(t)) dB(t) either stabilizes an unstable equilibrium, or destabilizes a stable equilibrium.
Almost sure polynomial asymptotic stability of stochastic difference equations
• Mathematics
• 2008
In this paper, we establish the almost sure asymptotic stability and decay results for solutions of an autonomous scalar difference equation with a nonhyperbolic equilibrium at the origin, which is
On local stability for a nonlinear difference equation with a non-hyperbolic equilibrium and fading stochastic perturbations
• Mathematics
• 2008
Here, (jn)n[N is a sequence of independent random variables with zero mean and unit variance and with distribution functions Fn. The function f : R ! R is continuous, f(0) 1⁄4 0, xf(x) . 0 for x – 0.

References

SHOWING 1-10 OF 15 REFERENCES
On boundedness and stability of solutions of nonlinear difference equation with nonmartingale type noise
• Mathematics
• 2001
Consider a stochastic difference equation with the Volterra type nonlinear main term G and Volterra type noise Functions Gf, σ supposed to be random, x;i is a martingale—difference. In general so Eq,
ON GLOBAL ASYMPTOTIC STABILITY OF SOLUTIONS OF SOME IN-ARITHMETIC-MEAN-SENSE MONOTONE STOCHASTIC DIFFERENCE EQUATIONS IN IR
• Mathematics
• 2005
Global almost sure asymptotic stability of the trivial solution of some nonlinear stochastic difference equations with in-the-arithmetic-meansense monotone drift part and diffusive part driven by
Global asymptotic stability of solutions of cubic stochastic difference equations
• Mathematics
• 2004
Global almost sure asymptotic stability of solutions of some nonlinear stochastic difference equations with cubic-type main part in their drift and diffusive part driven by square-integrable
On asymptotic behaviour of solutions of stochastic difference equations with volterra type main term
• Mathematics
• 2000
The stochastic difference equations with Volterra type linear and nonlinear main term are considered in the paper.Conditions on a.s. boundedness of the solutions,asymptotic stability,exponential
Stochastic self-stabilization
Given a nonlinear Ito equation suppose that the equation is almost surely exponentially stable provided the noise intensity u is large enough. We show that if u is replaced by then the equation
Brownian Motion and Stochastic Calculus
This chapter discusses Brownian motion, which is concerned with continuous, Square-Integrable Martingales, and the Stochastic Integration, which deals with the integration of continuous, local martingales into Markov processes.
Stabilization of Volterra equations by noise
• Mathematics
• 2002
The paper studies the stability of an autonomous convolution Ito-Volterra equation where the linear diffusion term depends on the current value of the state only, and the memory of the past fades