Dynamics of Fuzzy Bam Neural Networks with Distributed Delays and Diffusion

Copyright q 2012 Qianhong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Constructing a new Lyapunov functional and employing inequality technique, the existence, uniqueness, and global exponential stability of the periodic oscillatory solution are investigated for a class of fuzzy bidirectional associative memory BAM neural networks with distributed delays and diffusion. We obtained some sufficient conditions ensuring the existence, uniqueness, and global exponential stability of the periodic solution. The results remove the usual assumption that the activation functions are differentiable. An example is provided to show the effectiveness of our results.


Introduction
The bidirectional associative memory BAM neural network was first introduced by Kosko 1, 2 .These models generalize the single-layer autoassociative Hebbian correlator to a two layer pattern-matched heteroassociative circuits.BAM neural network is composed of neurons arranged in two-layers, the X-layer and the Y-layer.The BAM neural network has been used in many fields such as image processing, pattern recognition, and automatic control 3 .Recently, the stability and the periodic oscillatory solutions of BAM neural networks have been studied see, e.g., 1-25 .In 2002, Cao and Wang 7 derived some sufficient conditions for the global exponential stability and existence of periodic oscillatory solution of BAM neural networks with delays.Some authors 16, 19, 25 studied the BAM neural networks with distributed delays, which are more appropriate when neural networks have a multitude of parallel pathways with a variety of axon sizes and lengths.
However, strictly speaking, diffusion effects cannot be avoided in the neural networks when electrons are moving in asymmetric electromagnetic fields.So we must consider that activations vary in space as well as in time.Song et al. 25 have considered the stability of BAM neural networks with diffusion effects, which are expressed by partial differential equations.In this paper, we would like to integrate fuzzy operations into BAM neural networks.Speaking of fuzzy operations, T. Yang and L. B. Yang 26 first introduced fuzzy cellular neural networks FCNNs combining those operations with cellular neural networks.So far, researchers have founded that FCNNs are useful in image processing, and some results have been reported on stability and periodicity of FCNNs 26-32 .However, to the best of our knowledge, few authors consider the global exponential stability and existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion terms.In 33 , Li studied the global exponential stabilities of both the equilibrium point and the periodic solution for a class of BAM fuzzy neural networks with delays and reactiondiffusion terms.Motivated by the above discussion, in this paper, by constructing a suitable Lyapunov functional and employing inequality technique, we will derive some sufficient conditions of the global exponential stability and existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion terms.

System Description and Preliminaries
In this paper, we consider the globally exponentially stable and periodic fuzzy BAM neural networks with distributed delays and diffusion terms described by partial differential equations with delays: where n and m correspond to the number of neurons in X-layer and Y -layer, respectively.
x and v j t, x are the state of the ith neuron and the jth neurons at time t and in space x, respectively.a i > 0, b j > 0, and they denote the rate with which the ith neuron and jth neuron will reset its potential to the resting state in isolation when A2 The delay kernels K ji , N ij : 0, ∞ → 0, ∞ i 1, 2, . . ., n; j 1, 2, . . ., m are nonnegative continuous functions that satisfy the following conditions:

2.5
To be convenient, we introduce some notations.Let . ., v * m be the equilibrium of system 2.1 .We denote T of the delay fuzzy BAM neural networks 2.1 is said to be globally exponentially stable, if there exist positive constants for every solution u t, x , v t, x of the delay fuzzy BAM neural networks 2.1 with the initial conditions 2.2 and 2.3 for all t > 0.
Definition 2.2.If f t : R → R is a continuous function, then the upper right derivative of f t is defined as Lemma 2.3 see 26 .Suppose x and y are two states of system 2.1 , then one has 2.9 The remainder of this paper is organized as follows.In Section 3, we will study global exponential stability of fuzzy BAM neural networks 2.1 .In Section 4, we present the existence of periodic solution for fuzzy BAM neural networks 2.1 .In Section 5, an example will be given to illustrate effectiveness of our results obtained.We will give a general conclusion in Section 6.

Global Exponential Stability
In this section, we will discuss the global exponential stability of fuzzy BAM neural networks 2.1 by constructing suitable functional.Theorem 3.1.Under assumptions (A1) and (A2), if there exist δ i > 0, δ n j > 0 such that where i 1, 2, . . ., n, j 1, 2, . . ., m then the equilibrium u * , v * of system 2.1 is the globally exponentially stable.Proof.By using 3.1 , we can choose a small number λ > 0 such that N ij s e λs ds < 0, K ji s e λs ds < 0.

3.2
It is well known that the bounded functions always guarantee the existence of an equilibrium point for system 2.1 .The uniqueness of the equilibrium for system 2.1 will follow from the global exponential stability to be established below.Suppose u 1 t, x , . . ., u n t, x , v 1 t, x , . . ., v m t, x T is any solution of system 2.1 .Rewrite 2.1 as follows

3.4
Multiply both sides of 3.3 by u i − u * i and integrate, we have

3.5
Applying the boundary condition 2.2 and the Gauss formula, we get is the gradient operator and

3.8
Multiply both sides of 3.4 by v i − v * i , similarly, we get

3.9
We construct a Lyapunov functional

3.10
Calculating the upper right derivative D V t of V t along the solution of 3.3 and 3.4 , from 3.8 and 3.9 , we get

3.11
Estimating the right of 3.11 by elemental inequality 2ab ≤ a 2 b 2 , we obtain that

3.13
Noting that On the other hand, we have

3.15
Let 3.17 This implies that the equilibrium point of system 2.1 is globally exponentially stable.The proof is completed.
Corollary 3.2.Suppose (A1) and (A2) hold.Then, the equilibrium u * , v * of system 2.1 is the globally exponentially stable, if the following conditions are satisfied: 3.18

Periodic Oscillatory Solutions of Fuzzy BAM Neural Networks
In this section, we consider the existence and uniqueness of periodic oscillatory solutions for system 2.1 .

4.2
Journal of Applied Mathematics 13 For any Φ ∈ C, we define Then, C is the Banach space of continuous functions.
For any φ u , ϕ v T , φ u , ϕ v T ∈ C, we denote the solutions of system 2.1 through 0, 0 T , φ u , ϕ v T , and 0, 0 then u t φ u , x , v t ϕ v , x T φ u , x , v t ϕ v , x T ∈ C for t ≥ 0. Let u ix u i t, φ u , x − u i t, φ u , x , v jx v j t, ϕ v , x − v j t, ϕ v , x .Therefore, it follows from system 2.1 that

4.6
Considering the following Lyapunov functional

4.7
By a minor modification of the proof of Theorem 3.1, we can easily obtain

4.12
It implies that P N is a contraction mapping.Hence, there exist a unique equilibrium point

4.13
Note that x T is also the solution of system 2.1 .It is clear that for all t ≥ 0. It follows from 4.15 that which shows that u t, φ * u , x , v t, ϕ * v , x T is exactly an ω-periodic solution of system 2.1 with the initial conditions 2.2 and 2.3 and other solutions of system 2.1 with the initial conditions 2.2 and 2.3 converge exponentially to it as t → ∞.

An Illustrative Example
In this section, we will give an example to illustrate feasible our result.
Example 5.1.Consider the following system

5.4
Since all the conditions of Corollary 3.2 are satisfied, therefore the system 5.1 has a unique equilibrium point which is globally exponentially stable.

Conclusion
In this paper, we have studied the exponential stability of the equilibrium point and the existence of periodic solutions for fuzzy BAM neural networks with distributed delays and diffusion.Some sufficient conditions set up here are easily verified, and the conditions which are only correlated with parameters of the system 2.1 are independent of time delays.The results obtained in this paper remove the assumption about the activation functions with differentiable and only require the activation functions are bounded and Lipschitz continuous.Thus, it allows us even more flexibility in choosing activation functions.
disconnected from the network and external inputs; c ji and d ij are constants, denoting the connection weights.α ji , β ji , T ji , and H ji are elements of fuzzy feedback MIN template and fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feed-forward MAX template in X-layer, respectively; p ij , q ij , S ij , and L ij are elements of fuzzy feedback It is obvious that f • , g • satisfy assumption A1 and K • , N • satisfy assumption A2 , moreover μ 1 μ 2 ν 1 ν 2 1.Let