We introduce random towers to study almost sure rates of correlation decay for partially hyperbolic attractors. Using this framework, we obtain abstract results on exponential, stretched exponential and polynomial rates. then apply our small perturbations Axiom A attractors, derived from Anosov systems solenoidal attractors with intermittency.

This paper introduces a stochastic, hybrid algorithm for global almost sure synchronization of two agents evolving on the circle. Recent results on stochastic hybrid systems are exploited and a Lyapunov-based proof of global almost sure synchronization is obtained. The same arguments establish that global almost sure practical synchronization is achieved in the presence of sufficiently small di...

We study the random walk in random environment on Z+ = {0, 1, 2, . . .}, where the environment is subject to a vanishing (random) perturbation. The two particular cases that we consider are: (i) random walk in random environment perturbed from Sinai’s regime; (ii) simple random walk with random perturbation. We give almost sure results on how far the random walker is from the origin, for almost...

This paper is concerned with the almost sure partial practical stability of stochastic differential equations general decay rate. We establish some sufficient conditions based upon construction appropriate Lyapunov functions. Finally, we provide a numerical example to demonstrate efficiency obtained results.

and Applied Analysis 3 see 35 . When designing an associative memory neural network, we should make convergence speed as high as possible to ensure the quick convergence of the network operation. Therefore, pth moment p ≥ 2 exponential stability and almost sure exponential stability are most useful concepts as they imply that the solutions will tend to the trivial solution exponentially fast. T...

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 martingale differences is proven under appropriate conditions in R1. As an application of this result, the asymptotic stability of stochastic numerical methods, such as partially drift-implicit θ-methods wi...

This paper is concerned with the problem of almost sure state estimation for general nonlinear hybrid stochastic systems whose coefficients only satisfy local Lipschitz conditions. By utilizing the stopping time method combined with martingale inequalities, a theoretical framework is established for analyzing the so-called almost surely asymptotic stability of the addressed system. Within such ...