Stochastic Dynamics of Discrete-Time Fuzzy Random BAM Neural Networks with Time Delays
By using the semidiscrete method of differential equations, a new version of discrete analogue of stochastic fuzzy BAM neural networks was formulated, which gives a more accurate characterization for continuous-time stochastic neural networks than that by the Euler scheme. Firstly, the existence of the -th mean almost periodic sequence solution of the discrete-time stochastic fuzzy BAM neural networks is investigated with the help of Minkowski inequality, Hölder inequality, and Krasnoselskii’s fixed point theorem. Secondly, the -th moment global exponential stability of the discrete-time stochastic fuzzy BAM neural networks is also studied by using some analytical skills in stochastic theory. Finally, two examples with computer simulations are given to demonstrate that our results are feasible. The main results obtained in this paper are completely new, and the methods used in this paper provide a possible technique to study -th mean almost periodic sequence solution and -th moment global exponential stability of semidiscrete stochastic fuzzy models.
In [1, 2], Kosko introduced the bidirectional associative memory (BAM) neural networks, which have been widely applied in psychophysics, parallel computing, perception, robotics, adaptive pattern recognition, associative memory, image processing pattern recognition, combinatorial optimization, and so on. All of these applications heavily depend on the (almost) periodicity and global exponential stability [3–14]. In the past twenty years, a large, fast growing body of investigations focused on the existence and global exponential stability of the equilibrium point, periodic, and almost periodic solutions of BAM neural networks with time delays in literatures [15–27].
Fuzzy theory was conceived in the 1960s by L. A. Zadeh, and it took about 20 years until the broader use of this theory in practice. Fuzzy technology joined forces with artificial neural networks and genetic algorithms under the title of computational intelligence or soft computing. In recent years, the research on the dynamical behaviours of fuzzy neural networks has attracted much attention (see [28–33]). On the other hand, uncertain models described by the stochastic differential equations have caused great concern since uncertain models have widely applications in practice such as engineering, physics, chemistry, and biology [34–38]. In the actual situations, random factors have consequences on the performance of the neural networks. In neural networks, the connection weights of the neurons depend on certain resistance and capacitance values that include modeling errors or uncertainties during the parameter identification process. Therefore, it is worth studying the dynamical behaviors of fuzzy stochastic neural networks.
The discrete-time neural networks become more important than the continuous-time counterparts when implementing the neural networks in a digital way. In order to investigate the dynamical characteristics with respect to digital signal transmission, it is essential to formulate the discrete analogue of neural networks. Mohamad and Gopalsamy [39, 40] proposed a novel method (i.e., semidiscretization technique) in formulating a discrete-time analogue of the continuous-time neural networks, which faithfully preserved the characteristics of their continuous-time counterparts. With the help of the semidiscretization technique , many scholars obtained the semidiscrete analogue of the continuous-time neural networks and some meaningful results were gained for the dynamical behaviors of the semidiscrete neural networks, such as periodic solutions, almost periodic solutions and global exponential stability (see [41–47]).
For instance, Huang et al.  discussed the almost periodic dynamics of the following semidiscrete cellular neural networks:where , in which denotes the set of integers, and .
In , Huang et al. considered the following semidiscrete models for a class of general neural networks:where and . The authors  derived the existence of locally exponentially convergent almost periodic sequence solutions of system (2).
By careful observation, it easily discovers that the disquisitive models in literatures [41–47] are deterministic, such as (1) and (2). Stimulated by this point, it is necessary to consider random factors in the determinant models. Therefore, this paper considers the semidiscrete models for the following stochastic fuzzy BAM neural networks:
System (3) is composed of two layers, that is, X-layer and Y-layer. denotes the membrane potentials of the set of n neurons in X-layer, and denotes the membrane potentials of the set of m neurons in Y-layer at time t; and represent the measures of activation to its incoming potentials of the unit j from Y-layer and the unit i from X-layer, respectively; corresponds to the synaptic connection weight of the unit j on the unit i, and corresponds to the synaptic connection weight of the unit i on the unit j; and signify the bounded external bias or input; and denote rate with which the ith unit and jth unit will reset their potentials to the resting state in isolation when it is disconnected from the network and external inputs, respectively; , , , , , , and are elements of fuzzy feedback MIN, MAX template, fuzzy feed forward MIN, and MAX template, respectively; and denote the fuzzy AND and fuzzy OR operation, respectively; and are the standard Brownian motions, where and . The other coefficients in system (3) are similarly specified.
The main aim of this paper is to investigate the dynamics of the semidiscrete analogue of system (3) by using semidiscretization technique  and stochastic theory. The main contributions of this paper are summed up as follows: (1) the semidiscrete analogue is established for stochastic fuzzy BAM neural networks (3); (2) a Volterra additive equation is derived for the solution of the semidiscrete model; (3) the existence of -th mean almost periodic sequence solutions is obtained; (4) a decision theorem is acquired for the -th moment global exponential stability; and (5) the methods used in this article can be applied to study the dynamics of other discrete stochastic fuzzy models.
The work of this paper is a continuation of literatures [44–47], and the results in this paper complement the corresponding results in [44–47]. The paper is organized as follows. In Section 2, the discrete analogue of system (3) is established and some useful lemmas are given. In Section 3, we employ Krasnoselskii’s fixed point theorem to obtain sufficient conditions for the existence of at least one -th mean almost periodic sequence solution. In Section 4, we consider the -th moment global exponential stability. Two illustrative examples and computer simulations are given in Section 5. In Section 6, the conclusions and future works of this paper are presented.
Throughout this paper, we use the following notations. Let denote the set of real numbers. denotes the n-dimensional real vector space. Let be a complete probability space. Denote by the vector space of all bounded continuous functions from to , where denotes the collection of all p-th integrable -valued random variables. Then, is a Banach space with the norm , , , where and stands for the expectation operator with respect to the given probability measure P. , .
2. Model Formulation and Preliminaries
2.1. Discrete Analogue of System (3)
Consider the following stochastic functional differential equations:which yields the following stochastic functional differential equations with piecewise constant arguments:where denotes the integer part of t. Here, the discrete analogue of the stochastic parts of system (4) is obtained by the Euler scheme, i.e., . For , there exists an integer such that . Then, the above equation becomes
Integrating the above equation from k to t and letting , we achieve the discrete analogue of system (4) as follows:where . System (7) is the discrete analogue of system (4) by the semidiscrete method. By a similar discussion as that in system (7), we obtain the semidiscrete analogue of system (3) as follows:where and .
2.2. Volterra Additive Equation for the Solution of System (8)
Lemma 1. Assume that is a solution of system (8), and then X can be expressed aswhere , , , and .
Proof. LetBy and the first equation of system (8), it getswhere and . So,is equivalent towhere and . By the above equations, we can easily derive the first equation of system (9). Similarly, we can obtain the second equation of system (9). This completes the proof.
2.3. Some Lemmas
Lemma 2 (Minkowski inequality) . Assume that , , , and then
Lemma 3 (Hölder inequality) . Assume that , and thenIf , then .
Lemma 4 . Suppose that , and thenwhere
Lemma 5. Assume that is real-valued stochastic process and is the standard Brownian motion, and thenwhere is defined as that in Lemma 4, in which .
Proof. By Lemma 4, it follows thatThis completes the proof.
Lemma 7 . Assume that is a closed convex nonempty subset of a Banach space . Suppose further that and map into such that (1) implies that (2) is continuous and is contained in a compact set (3) is a contraction mappingThen, there exists a with .
Set and for bounded function f defined on . Throughout this paper, suppose that the following conditions are satisfied: and are bounded sequences defined on with and , , and . There exists several constants , , , and such that for all , , and .
3. 2p-th Mean Almost Periodic Sequence Solution
Definition 3.1 . A stochastic process is said to be p-th mean almost periodic sequence if for each , there exists an integer such that each interval of length contains an integer ω for which
A stochastic process X, which is 2-nd mean almost periodic sequence, is called square-mean almost periodic sequence. Like for classical almost periodic functions, the number ω is called an ϵ-translation of X.
Theorem 1. Assume that all of the coefficients in system (8) are almost periodic sequences, and then – hold and the following condition is satisfied: , where Then, there exists a -th mean almost periodic sequence solution X of system (8) with .
Proof. Let be the collection of all -th mean almost periodic sequences X satisfying the inequality .
Define , wherewhere , , and .
Let be defined aswhere , , and . By Minkoswki inequality in Lemma 2 and Hölder inequality in Lemma 3, we haveFrom Lemma 6, it gets from the above inequality thatSimilarly, we haveBy the above inequalities, it concludesIt follows (25)–(27) thatwhich yields from Lemmas 2 and 3 thatwhere , . Applying Lemma 5 to the above inequality, it derivesSimilarly,Together with the above inequalities, we obtainHence, for , it leads from (32) and (37) toFrom (38), is uniformly bounded. Together with the continuity of , for any bounded sequence in , we know that there exists a subsequence in such that is convergent in . Therefore, is compact on . Then, condition (2) of Lemma 7 is satisfied.
The next step is proving condition (1) of Lemma 7. Now, we consist in proving the -th mean almost periodicity of and . Since is a -th mean almost periodic sequence and all the coefficients in system (8) are almost periodic sequences, for all there exists such that every interval of length contains a ω with the property thatwhere , , and . By (26), (27), and , we could easily find a positive constant M such that