scholarly journals Convergence and Stability of the Split-Stepθ-Milstein Method for Stochastic Delay Hopfield Neural Networks

2013 ◽  
Vol 2013 ◽  
pp. 1-12
Author(s):  
Qian Guo ◽  
Wenwen Xie ◽  
Taketomo Mitsui
Keyword(s):  
Neural Networks ◽  
Linear Growth ◽  
Numerical Solutions ◽  
Splitting Method ◽  
Growth Conditions ◽  
Hopfield Neural Networks ◽  
Mean Square ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Milstein Method

A new splitting method designed for the numerical solutions of stochastic delay Hopfield neural networks is introduced and analysed. Under Lipschitz and linear growth conditions, this split-stepθ-Milstein method is proved to have a strong convergence of order 1 in mean-square sense, which is higher than that of existing split-stepθ-method. Further, mean-square stability of the proposed method is investigated. Numerical experiments and comparisons with existing methods illustrate the computational efficiency of our method.

Exponential Stability of Numerical Solutions to a Stochastic Delay Neural Networks

2011 ◽  
Vol 219-220 ◽  
pp. 1035-1039
Author(s):  
Qi Min Zhang
Keyword(s):  
Neural Networks ◽  
Numerical Solutions ◽  
Euler Scheme ◽  
Mean Square ◽  
Approximation Solution ◽  
True Solution ◽  
Stochastic Delay ◽  
Mean Square Stability ◽  
Exponentially Stable ◽  
Definition Of

The main purpose of this paper is to develop a numerical Euler scheme and show the convergence of the numerical approximation solution to the true solution for stochastic delay neural networks. The definition of exponential mean square stability of numerical method is introduced. It is proved that the Euler scheme is exponentially stable in mean square sense. An example is given for illustration our result.

scholarly journals Stability of Analytical and Numerical Solutions for Nonlinear Stochastic Delay Differential Equations with Jumps

2012 ◽  
Vol 2012 ◽  
pp. 1-13 ◽  
Author(s):  
Qiyong Li ◽  
Siqing Gan
Keyword(s):  
Exact Solution ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Solutions ◽  
Mean Square ◽  
Stochastic Delay Differential Equations ◽  
Stochastic Delay ◽  
Analytical And Numerical Solutions ◽  
Mean Square Stability ◽  
Delay Differential

This paper is concerned with the stability of analytical and numerical solutions fornonlinearstochastic delay differential equations (SDDEs) with jumps. A sufficient condition for mean-square exponential stability of the exact solution is derived. Then, mean-square stability of the numerical solution is investigated. It is shown that the compensated stochastic θ methods inherit stability property of the exact solution. More precisely, the methods are mean-square stable for any stepsizeΔt=τ/mwhen1/2≤θ≤1, and they are exponentially mean-square stable if the stepsizeΔt∈(0,Δt0)when0≤θ<1. Finally, some numerical experiments are given to illustrate the theoretical results.

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