Dynamic analysis of stochastic bidirectional associative memory neural networks with delays

doi:10.1016/j.chaos.2005.12.010

Chaos, Solitons & Fractals

Volume 32, Issue 5, June 2007, Pages 1692-1702

https://doi.org/10.1016/j.chaos.2005.12.010 Get rights and content

Abstract

In this paper, stochastic bidirectional associative memory neural networks model with delays is considered. By constructing Lyapunov functionals, and using stochastic analysis method and inequality technique, we give some sufficient criteria ensuring almost sure exponential stability, pth exponential stability and mean value exponential stability. The obtained criteria can be used as theoretic guidance to stabilize neural networks in practical applications when stochastic noise is taken into consideration.

Introduction

Recently, the dynamics such as stability and periodicity of bidirectional associative memory (BAM) neural networks have received much attention due to their potential application in associative memory, parallel computation and optimization problems. Some important results have been obtained in Refs. [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Most neural network models proposed and discussed in the literature are deterministic. As is well known, a real system is usually affected by external perturbations which in many cases are of great uncertainty and hence may be treated as random, as pointed out by Haykin [12] that in real nervous system synaptic transmission is a noisy process brought on by random fluctuations from the release of neurotransmitters and other probabilistic causes. Therefore, it is of prime importance and great interest to consider stochastic effects to the stability of neural networks. To date, some results on stability of stochastic cellular neural networks and stochastic Cohen–Grossberg neural networks have been reported (see, [13], [14], [15], [16], [17]). However, to the best of our knowledge, few authors study the stability of stochastic BAM neural networks with delays.

Motivated by the above discussion, in this paper, we analyze the stochastic BAM neural networks model with delays. By constructing Lyapunov functional, and using stochastic analysis method and inequality technique, we give a set of sufficient conditions ensuring almost sure exponential stability, pth exponential stability and mean value exponential stability of an equilibrium point. The obtained criteria can be used as theoretic guidance to stabilize neural networks in practical applications when stochastic noise is taken into consideration.

Section snippets

Preliminary

Consider stochastic BAM neural networks model with delays $\{\begin{matrix} d u_{i} (t) = [- c_{i} u_{i} (t) + \sum_{j = 1}^{n} p_{ji} f_{j} (v_{j} (t - τ_{j})) + I_{i}] d t + \sum_{j = 1}^{n} σ_{ji} (v_{j} (t)) d ω_{j} (t), \\ d v_{j} (t) = [- d_{j} v_{j} (t) + \sum_{i = 1}^{n} q_{ij} g_{i} (u_{i} (t - η_{i})) + J_{j}] d t + m_{i = 1}^{n} σ_{ij} (u_{i} (t)) d ω_{i} (t), t ⩾ 0, \\ u_{i} (t) = ξ_{i} (t), v_{j} (t) = ζ_{j} (t), - τ ⩽ t ⩽ 0, \end{matrix}$ in which i, j = 1, … , n; 0 ⩽ τ_j, η_i ⩽ τ; u_i(t), v_j(t) denote the potential of the cell i and j at time t, respectively; c_i, d_j are positive constants, they denote the rate with which the cell i and j reset their potential to the resting state when isolated from the other cells and

Almost sure exponential stability

Theorem 1

If (A1) and (A2) hold, then system (3) is almost surely exponentially stable.

Proof

It follows from (A2) that there exists a sufficiently small constant 0 < c < min{c, d} such that $- r_{i} (c_{i} - c) + \sum_{j = 1}^{n} s_{j} | q_{ij} |^{r_{ij}^{*}} β_{i}^{s_{ij}^{*}} e^{c τ} + \frac{(q - 1) (q - 2)}{2} r_{i} \sum_{j = 1}^{n} L_{ji}^{\frac{2 b_{ij}}{q - 2}} < 0$ and $- s_{j} (d_{j} - c) + \sum_{i = 1}^{n} r_{i} | p_{ji} |^{r_{ij}} α_{j}^{s_{ij}} e^{c τ} + \frac{(q - 1) (q - 2)}{2} \sum_{i = 1}^{n} s_{j} L_{ij}^{\frac{2 b_{ij}^{*}}{q - 2}} < 0 .$

Taking $V (t) = \frac{1}{q} e^{ct} (\sum_{i = 1}^{n} r_{i} | y_{i} (t) |^{q} + \sum_{j = 1}^{n} s_{j} | z_{j} (t) |^{q})$ , and applying Itô’s formula to V(t), we have $V (t) = V (0) + \int_{0}^{t} \frac{1}{q} c e^{cs} \sum_{i = 1}^{n} r_{i} | y_{i} (s) |^{q} d s + \int_{0}^{t} \frac{1}{q} c e^{cs} \sum_{j = 1}^{n} s_{j} | z_{j} (s) |^{q} d s + \int_{0}^{t} e^{cs} \sum_{i = 1}^{n} r_{i} | y_{i} (s) |^{q - 1} sgn (y_{i} (t)) \{- c_{i} (u_{i}\}$

Moment exponential stability

Theorem 2

If (A1) and (A2) hold, then system (3) is pth exponentially stable.

Proof

Let $c^{*} = \frac{qc}{p}$ , by Theorem 1 we have $\min_{1 ⩽ i ⩽ n, 1 ⩽ j ⩽ n} {r_{i}, s_{j}} e^{c^{*} t} (\sum_{i = 1}^{n} | y_{i} (t) |^{q} + \sum_{j = 1}^{n} | z_{j} (t) |^{q}) ⩽ q \sum_{i = 1}^{n} r_{i} | y_{i} (0) |^{q} + q \sum_{j = 1}^{n} s_{j} | z_{j} (0) |^{q} + \int_{- τ}^{0} e^{c^{*} s} \sum_{i = 1}^{n} \sum_{j = 1}^{n} r_{i} | p_{ji} |^{r_{ij}} α_{j}^{s_{ij}} e^{c^{*} τ} | z_{j} (s) |^{q} d s + \int_{- τ}^{0} e^{c^{*} s} \sum_{i = 1}^{n} \sum_{j = 1}^{n} s_{j} | q_{ij} |^{r_{ij}^{*}} β_{i}^{s_{ij}^{*}} e^{c^{*} τ} | y_{i} (s) |^{q} d s + q \int_{0}^{t} e^{c^{*} s} \sum_{i = 1}^{n} \sum_{j = 1}^{n} r_{i} | y_{i} (s) |^{q - 1} σ_{ji} (v_{j} (s)) d ω_{j} (s) + q \int_{0}^{t} e^{c^{*} s} \sum_{i = 1}^{n} \sum_{j = 1}^{n} s_{j} | z_{j} (s) |^{q - 1} σ_{ij} (u_{i} (s)) d ω_{i} (s) .$ From (A2), we have $\min_{1 ⩽ i ⩽ n, 1 ⩽ j ⩽ n} {r_{i}, s_{j}} e^{c^{*} t} ‖ (y (t), z (t))^{T} ‖_{q}^{q} ⩽ q \max_{1 ⩽ i ⩽ n, 1 ⩽ j ⩽ n} {r_{i}, s_{j}} ‖ (ϕ, ψ)^{T} ‖_{q}^{q} + q \int_{0}^{t} e^{c^{*} s} \sum_{i = 1}^{n} \sum_{j = 1}^{n} r_{i} | y_{i} (s) |^{q}$

Examples

Example 1

Consider a stochastic BAM neural networks with delays $\{\begin{matrix} d u_{1} (t) = - c_{1} u_{1} (t) d t + p_{11} f_{1} (v_{1} (t - τ_{1})) d t + σ_{11} (v_{1} (t)) d ω_{1} (t), \\ d v_{1} (t) = - d_{1} v_{1} (t) d t + q_{11} g_{1} (u_{1} (t - η_{1})) d t + σ_{11} (u_{1} (t)) d ω_{1} (t), \end{matrix}$ where f₁(v₁) = v₁, $σ_{11} (v_{1}) = v_{1}, g_{1} (u_{1}) = \frac{e^{u_{1}} - e^{- u_{1}}}{e^{u_{1}} + - e^{- u_{1}}}$ , σ₁₁(u₁) = u₁. Obviously, f₁, g₁, σ₁₁(v₁) satisfy the conditions (A1) with α₁ = β₁ = L₁₁ = 1. By taking p₁₁ = q₁₁ = 1, c₁ = 5, d₁ = 10, $q = 3, r_{1} = 3, s_{1} = 2, r_{11} = r_{11}^{*} = 1, s_{11} = s_{11}^{*} = 2, b_{11} = b_{11}^{*} = 1$ , it is very easy to see the condition (A2) holds. It follows Theorem 1, Theorem 2 that system (7) is almost surely

Conclusions

In this paper, stochastic bidirectional associative memory neural networks with delays is studied. By constructing Lyapunov functionals, and using stochastic analysis method and inequality technique, we obtain some sufficient conditions of the almost sure exponential stability, pth exponential stability and mean value exponential stability.The obtained conditions can be used as theoretic guidance to stabilize neural networks in practical applications when stochastic noise is taken into

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^☆: This research was supported by the National Natural Science Foundation of Nanjing University of Aeronautics and Astronautics.

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Dynamic analysis of stochastic bidirectional associative memory neural networks with delays☆

Abstract

Introduction

Section snippets

Preliminary

Almost sure exponential stability

Moment exponential stability

Examples

Conclusions

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Chaos, Solitons & Fractals

Appl Math Comput

Appl Math Comput

Phys Lett A

J Franklin Inst

Neural Networks