Stochastic Dynamics of Discrete-Time Fuzzy Random BAM Neural Networks with Time Delays

Han, Sufang; Zhang, Tianwei; Liu, Guoxin

doi:https://doi.org/10.1155/2019/9416234

Mathematical Problems in Engineering

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Abstract Introduction Model Formulation and Preliminaries Examples Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2019 | Article ID 9416234 | https://doi.org/10.1155/2019/9416234

Stochastic Dynamics of Discrete-Time Fuzzy Random BAM Neural Networks with Time Delays

Sufang Han,¹Tianwei Zhang,²and Guoxin Liu¹

Academic Editor: Jose Vicente Salcedo

Received30 Apr 2019

Revised20 Jul 2019

Accepted04 Aug 2019

Published25 Aug 2019

Abstract

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.

1. Introduction

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 [39], 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. [44] discussed the almost periodic dynamics of the following semidiscrete cellular neural networks:where , in which denotes the set of integers, and .

In [45], Huang et al. considered the following semidiscrete models for a class of general neural networks:where and . The authors [45] 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 [39] 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) [48]. Assume that , , , and then

Lemma 3 (Hölder inequality) [48]. Assume that , and thenIf , then .

Lemma 4 [35]. 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 6 [49]. Suppose and are two states of system (8), and then we have

Lemma 7 [50]. 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

Definewhere

Definition 3.1 [34]. 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 thatwhere , , and . From (40) and (41), and are -th mean almost periodic processes. Further, by (38), it is easy to obtain that , . Then, condition (1) of Lemma 7 holds.
Finally, , from (27), it yieldssimilarly,From the above inequalities, it leadsIn view of , is a contraction mapping. Hence, condition (3) of Lemma 7 is satisfied. Therefore, all the conditions in Lemma 7 hold. By Lemma 7, system (8) has a -th mean almost periodic sequence solution. This completes the proof.

According to Theorem 1, we can easily obtain the following theorem.

Theorem 2. Assume that all conditions in Theorem 1 hold and all coefficients in system (8) are periodic sequences, and then system (8) admits at least one -th mean periodic sequence solution.

Remark 1. In view of the definition of in Theorem 1, and in system (8) are bigger, then the probability of is bigger, where , and . The parameters , , , , , , , and have no effect on the value of , but they have effects on the boundedness of almost periodic sequence solutions of system (8), where and . In order to derive the existence of almost periodic sequence solutions of system (8), we would better choose bigger and , and the other parameters should be smaller except , , , , , , , and , where and .

4. 2p-th Moment Global Exponential Stability

Suppose that with initial value and with initial value are arbitrary two solutions of system (8). For convenience, let

Definition 4.1 [35]. System (8) is said to be -th moment global exponential stability if there are positive constants , M, and λ such thatwhere . The 2-nd moment global exponential stability is called square-mean global exponential stability.

Theorem 3. Assume that – hold, and then system (8) is -th moment globally exponentially stable.

Proof. From Lemmas 1 and 6, it follows thatwhere , , and .
For convenience, let, , , and , where , , and .
Similar to the argument as that in (37), it gets from (47) thatSimilarly, it follows from (48) thatBe aware of in Theorem 1, there exists a constant small enough such thatDefine , .
Next, we claim that there exists a constant such thatIf (53) is invalid, then there must exist such that one of the following two cases holds: and and If is valid, it follows from (50) thatIn the fourth inequality from the bottom of (54), we use the fact and . (54) contradicts . So, is invalid. Similarly, we can easily conclude is invalid by using (51). Therefore, (53) is satisfied. By (53), system (8) is -th moment globally exponentially stable. This completes the proof.

According to Theorems 1 and 2, we can easily obtain the following theorem.

Theorem 4. Assume that all conditions in Theorem 1 hold, and then system (8) admits a -th mean almost periodic sequence solution, which is -th moment globally exponentially stable. Further, if all coefficients in system (8) are periodic sequences, then system (8) admits at least one -th mean periodic sequence solution, which is globally exponentially stable.

Proof. The result can be easily obtained by Theorem 3, so we omit it. This completes the proof.

Remark 2. Assume that is a solution of (1), and then the length of is usually measured by . However, if is a solution of stochastic system, its length should not be measured by because is a random variable. In this paper, we use norm for random variable . Owing to the expectation E and order p in , the computing processes of this paper are more complicated than those in literatures [41–47]. It is worth mentioning that Minkoswki inequality in Lemma 2 and Hölder inequality in Lemma 3 are crucial to the computing processes. The facts above are obvious from the computations of (32), (42), (50), and (54) in the proofs of Theorems 1 and 3. Further, the stochastic terms and in system (8) also increase the complexity of computing. This point is also clear from the computations of (42) and (50).

5. Examples and Computer Simulations

Example 5.1. Consider the following continuous-time BAM neural networks with random perturbation:

5.1. Semidiscrete Model

Based on model (55), we obtain the following semidiscrete model by using the semidiscretization technique:where .

Remark 3. In literature [44], Huang et al. studied model (1) and obtained some sufficient conditions for the existence of a unique almost periodic sequence solution which is globally attractive. In [45], they considered system (2) and studied the dynamics of almost periodic sequence solutions. But neither of them considered the random factors. Such as, the obtained results in [44, 45] cannot be applied to the study for stochastic model (56). Therefore, the work in this paper complements the corresponding results in [44, 45].

5.2. Discrete Model Formulated by the Euler Scheme

Based on model (55), we obtain the following discrete-time model by using the Euler method:where .

In Figures 1 and 2, we give two plots of numerical solutions which are produced by continuous-time model (55), semidiscrete model (56), and Euler-discretization model (57), respectively. Compared with Euler-discretization model (57), semidiscrete model (56) gives a more accurate characterization for continuous-time model (55).

Remark 4. In literature [51–54], the authors discussed the dynamics of periodic solutions of discrete-time cellular neural networks formulated by the Euler scheme. From the above discussion, semidiscrete stochastic system (8) gives a more accurate and realistic formulation for studying the dynamics of discrete-time cellular neural networks. In a way, the work of this paper complements and improves some corresponding results in [51–54].
Corresponding to system (8), we have , , , , , , , , , , and .
Taking , by simple calculation,According to Theorems 1 and 3, system (56) admits a 4-th mean almost periodic sequence solution, which is 4-th moment globally exponentially stable (see Figures 3–5).
Figure 3 depicts a numerical solution of semidiscrete stochastic model (56). Observe that the trajectories of demonstrate almost periodic oscillations. Figures 4 and 5 display three numerical solutions of semidiscrete stochastic model (56) at different initial values , , and , respectively. They are shown that semidiscrete stochastic model (56) is 4-th moment globally exponentially stable.
Example 5.2. Consider the corresponding determinant model of system (56) as follows:where . In Figures 6 and 7, we give the results of contrast between stochastic model (56) and determinant model (59). Figures 6 and 7 indicate that the effects of stochastic perturbation on state variables x and y are significant. And the stochastic influence on state variable y is more obvious than that on state variable x. In Figure 8, we give a result of globally exponentially stable contrast between stochastic model (56) and determinant model (59). Figure 8 reveals that the convergence speed of stochastic model (56) is faster than determinant model (59).

(a)

(b)

Remark 5. From example 5.2, stochastic disturbance brings a positive effect on the global exponential stability of the models.

6. Conclusions and Future Works

In this paper, we formulate a discrete analogue of BAM neural networks with stochastic perturbations and fuzzy operations by using semidiscretization technique. The existence of -th mean almost periodic sequence solutions and -th moment global exponential stability for the above models are investigated with the help of Krasnoselskii’s fixed point theorem and stochastic theory. 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 models with stochastic perturbations and fuzzy operations.

With a careful observation of Theorems 1 and 3, it is not difficult to discover that(1) is crucial to the -th mean almost periodicity and moment global exponential stability of system (8)(2)The time delays have no effect on the existence of -th mean almost periodicity and -th moment global exponential stability of system (8)(3)Stochastic disturbance may bring a positive effect on the global exponential stability for determinant model

In the future, the following aspects can be explored further:(1)The methods used in this paper can be applied to study other types of neural networks, such as impulsive neural networks, high-order neural networks, neural networks on time scales, etc.(2)Other types of fuzzy neural networks could be investigated, such as Takagi-Sugeno fuzzy neural networks(3)Other dynamic behaviours of system (8) should be further discussed(4)The case of could be further explored

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the National Nature Science Foundation (nos. 11461082, 11601474, and 61472093) and Key Laboratory of Numerical Simulation of Sichuan Province (no. 2017KF002).

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Copyright © 2019 Sufang Han 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.

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