Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays

doi:10.1016/j.amc.2005.01.029

Applied Mathematics and Computation

Volume 171, Issue 1, 1 December 2005, Pages 108-121

https://doi.org/10.1016/j.amc.2005.01.029 Get rights and content

Abstract

In this paper, we study the periodic oscillatory behavior of a class of bidirectional associative memory (BAM) networks with finite distributed delays. A set of criteria are proposed for determining global exponential periodicity of the proposed BAM networks, which assume neither differentiability nor monotonicity of the activation function of each neuron. In addition, our criteria are easily checkable.

Introduction

Bidirectional associative memory (BAM) networks are a class of important neural network with the ability to store a collection of pattern pairs via unsupervised learning, which have applications in pattern recognition and automatic control [12], [13]. Recently, the stability properties of BAM networks have been extensively studied [2], [3], [4], [5], [6], [7], [8], [9], [10], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29].

Time delays are often encountered in neural network models. They are the source of oscillation and instability [1], [11]. BAM models with (fixed or time-varying) discrete time delays provide a good approximation to simple circuits consisting of a small number of neurons [2], [3], [4], [5], [6], [7], [8], [10], [16], [17], [18], [20], [22], [23], [24], [27], [28]. Real BAM networks, however, usually have a spatial extent due to the presence of a multitude of parallel pathways with a variety of axon sizes and lengths, which may result in distributed transmission delays. Recently, BAM networks with infinite distributed delays have been deeply studied [9], [14], [15], [19], [21], [25], [26], [29].

It is well known that existence and stability of periodic solution to a BAM network play an important role in applications such as associative memory and repetitive learning. Various results on global asymptotic/exponential periodicity of BAM networks with discrete delays were reported in literature [2], [5], [10], [22], [23], [24], [27]. Very recently, Li [14] studied global asymptotic periodicity of a class of BAM networks with infinite distributed delays; while Chen et al. [7] and Liu et al. [21] reported some interesting results on existence and stability of almost periodic solution to a class of discrete delayed BAM networks and a class of BAM networks with infinite distributed delays, respectively.

It should be noted that a distributed delayed BAM network with compactly supported delay kernels will reduce to a BAM network with finite distributed delays. The study of finite distributed delayed BAM networks not only is valuable in its own right, but the resulting conclusions can provide insight into their counterparts with infinite distributed delays. To our knowledge, few results on BAM networks with finite distributed delays have been reported in literature.

This paper addresses global exponential periodicity of a class of BAM networks with finite distributed delays. A set of criteria for determining global exponential periodicity of the indicated network are derived. Our criteria assume neither differentiability nor monotonicity of the activation function of each neuron. In addition, these criteria are easily checkable.

The materials are organized as follows: In Section 2, the BAM network model under consideration is formulated, and the preliminary knowledge is provided. The main results are established in Section 3, while an illustrative example is given in Section 4 to show the validity of our criteria. Some conclusions are drawn in Section 5.

Section snippets

Preliminaries

In this paper we will investigate the following BAM network with finite distributed delays $\{\begin{matrix} {\dot{x}}_{i} (t) = - a_{i} x_{i} (t) + \sum_{j = 1}^{q} \int_{0}^{τ} b_{ij} (s) g_{j} (y_{j} (t - s)) d s + I_{i} (t), i = 1, 2, \dots, p; \\ {\dot{y}}_{j} (t) = - c_{j} y_{j} (t) + \sum_{i = 1}^{p} \int_{0}^{τ} d_{ji} (s) h_{i} (x_{i} (t - s)) d s + J_{j} (t), j = 1, 2, \dots, q; \end{matrix}$ with the initial values $(\begin{matrix} x (θ) \\ y (θ) \end{matrix}) = ϕ (θ) for θ \in [- τ, 0] .$ Here, p and q are the respective numbers of neurons in the first and second layers of the indicated BAM network, x(t) = (x₁(t), … , x_p(t))^T and y(t) = (y₁(t), … , y_q(t))^T are the respective state vectors of the first and second layers at time t, h(x(t

Main results

This section aims at establishing some sufficient conditions for determining global exponential periodicity of system (2.1).

Theorem 3.1

Suppose there exist positive numbers λ₁, λ₂, … , λ_p+q such that $\{\begin{matrix} λ_{i} (2 a_{i} - \sum_{j = 1}^{q} L_{j}^{(2)} \int_{0}^{τ} ∣ b_{ij} (s) ∣ d s) > L_{i}^{(1)} \sum_{j = 1}^{q} λ_{p + j} \int_{0}^{τ} ∣ d_{ji} (s) ∣ d s, i = 1, 2, \dots, p; \\ λ_{p + j} (2 c_{j} - \sum_{i = 1}^{p} L_{i}^{(1)} \int_{0}^{τ} ∣ d_{ji} (s) ∣ d s) > L_{j}^{(2)} \sum_{i = 1}^{p} λ_{i} \int_{0}^{τ} ∣ b_{ij} (s) ∣ d s, j = 1, 2, \dots, q . \end{matrix}$ Then system (2.1) is globally exponentially periodic.

To prove the above result, we need the following lemma.

Lemma 3.2

Let $(\begin{matrix} x (t, ϕ) \\ y (t, ϕ) \end{matrix})$ , $(\begin{matrix} x (t, ϕ) \\ y (t, ϕ) \end{matrix})$ be a pair of solutions to (2.1). If there

An illustrative example

Consider the system $\{\begin{matrix} {\dot{x}}_{i} (t) = - a_{i} x_{i} (t) + \sum_{j = 1}^{q} \int_{0}^{τ} b_{ij} (s) g_{j} (y_{j} (t - s)) d s + I_{i} (t), i = 1, 2, \dots, p; \\ {\dot{y}}_{j} (t) = - c_{j} y_{j} (t) + \sum_{i = 1}^{p} \int_{0}^{τ} d_{ji} (s) h_{i} (x_{i} (t - s)) d s + J_{j} (t), j = 1, 2, \dots, q; \end{matrix}$ with initial conditions ϕ ∈ C. Here h_i(•) = g_j(•) = tanh(•), I_i(•) and J_j(•) are ω-periodic continuous functions, b_ij(•) = d_ji(•) = r(1 − s/τ) (r > 0). Then $L_{i}^{(1)} = L_{j}^{(2)} = 1$ . When $\min_{1 ⩽ i ⩽ p} {a_{i}} > \frac{1}{2} rq τ, \min_{1 ⩽ j ⩽ q} {c_{j}} > \frac{1}{2} rp τ,$ it follows from Corollary 3.3 that system (4.1) is globally exponentially periodic; while when $\min_{1 ⩽ i ⩽ p} {a_{i}} > \frac{1}{6} (r^{2} + 3) q τ, \min_{1 ⩽ j \leq q} {c_{j}} > \frac{1}{6} (r^{2} + 3) p τ,$ we conclude from

Conclusions

BAM networks with finite distributed delays are good approximations to BAM networks with infinite distributed delays. A set of sufficient conditions have been derived for global exponential periodicity of BAM networks with finite distributed delays, which are easily checkable. Our results are instructive in the design of globally exponentially periodic BAM networks.

Acknowledgments

This work is supported jointly by Chinese National Natural Science Funds (60271019) and Chongqing’s Application-Oriented Fundamentals Research Funds (8028).

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Global exponential periodicity of a class of bidirectional associative memory networks with finite distributed delays

Abstract

Introduction

Section snippets

Preliminaries

Main results

An illustrative example

Conclusions

Acknowledgments

Appl. Math. Comput.

Appl. Math. Comput.

Phys. Lett. A

Neurocomputing

Appl. Math. Comput.

Phys. Lett. A

Appl. Math. Comput.

Appl. Math. Comput.

Chaos Solitons Fract.

Phys. Lett. A

Phys. Lett. A

J. Math. Anal. Appl.

Phys. Lett. A

Chaos Solitons Fract.