Robustness Analysis of BAM Cellular Neural Network with Deviating Arguments of Generalized Type

Fang, Wenxiang; Xie, Tao; Li, Biwen

doi:https://doi.org/10.1155/2023/9570805

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 9570805 | https://doi.org/10.1155/2023/9570805

Robustness Analysis of BAM Cellular Neural Network with Deviating Arguments of Generalized Type

Wenxiang Fang,¹Tao Xie,¹and Biwen Li¹

Academic Editor: Junwei Wang

Received06 Sept 2022

Revised03 Feb 2023

Accepted10 Mar 2023

Published22 Apr 2023

Abstract

By generating equivalent integral equations, we analyze the existence and uniqueness of solutions of bidirectional associative memory cellular neural network (BAMCNN) with deviating arguments firstly. Secondly, the question of robustness of stability (RoS) of BAMCNN with deviating argument is studied. Using the Gronwall inequality, we calculate the upper bounds of the interference intensities that can maintain the initial stability of system. The perturbed BAMCNN will maintain its original stability if the strength of one or more perturbations is less than the upper bounds that we calculated in this study. To demonstrate the validity of the conjectural values, a variety of numerical illustrations are provided.

1. Introduction

As a kind of nonlinear dynamic system with structure law and infinitely expandable dimension, cellular neural network (CNN) is first proposed by Chua and Yang in 1988 [1]. Cells are the basic architecture of CNN which are essentially similar analog circuits, grouped in any arrays which dimensions larger than 2. CNNs have adjacency, it is one of the most significant properties of those systems, that is, every unit connects with its neighbors in the same way. Its dynamics primarily exhibit chaotic, periodic, almost periodic, and stable properties. We can refer to [2, 3] and any related references.

Bidirectional associative memory cellular neural network (BAMCNN) as a model of supervised learning in artificial neural network, were first proposed by Kosko in [4] in 1988. In application, it needs recurrent neural networks to receive a set of neurons as an input and create a set of outputs that related but different with inputs. Besides, the X-layer and the Y-layer are the two layers of BAMCNNs. Neurons in the same layer are completely linked to those in the other layer, and there are no connections between neurons in one layer. And its dynamical behaviors gain more and more interests in recent decades because of its special properties and usefulness in several domains, such as image identification, the problem of optimization, and other areas [5, 6].

Stability as one of the most important dynamical behaviors has received more extensive attention [7–16]. For example, the global exponential stability (ES) of periodic solutions of delayed BAMNN (DBAMNN) of cohen-crossberg type in [7]. In [8], the ES of delayed impulsive discrete-time stochastic BAMNN (SBAM) is investigated by using Lyapunov theory and contraction mapping principle. Several criteria of delay-dependent asymptotical stability (AS) of uncertain BAMNN are derived in [9]. Liu et al. explore the stabilization of uncertain BAMNN in finite time in [10]. In [11], Chen and Cao et al. discarded the common assumptions and some analytic techniques are used to estimate the exponential convergence rate of DBAMNN. By constructing Lyapunov functions, the ES of stochastic DBAMNN is studied in [12]. AS of a class of SBAMs is investigated in [13]. In [15, 16], Zhu et al.'s further analysis of the stability of delayed SBAM with Markovian jumping is conducted.

It is worth mentioning that all the above researches are all devoted to the stability of BAMNN without deviating argument. However, for some complex physical problems, such as the problem of population dynamics [17], it is affected not only by the delayed states but also by the advanced states. The potential of anticipation in systems may be viewed as a result of the analytical model’s complexity. This additional complexity is discovered by treating the time dimension as a degree of freedom accessible to the system rather than as a given. Thus, neural networks with time delays or stochastic disturbances cannot simulate this desired outcome well, so the deviating argument theory is introduced. Differential equation with deviating argument is a hybrid of continuous and discrete equation that combines the traits of differential and difference equations. Meanwhile, noise disturbances are inevitable in actual nervous systems. All of these disturbances have a significant impact on the stability of dynamical systems. Consequently, it is essential to take into account stochastic neural networks with deviating arguments. And many results of neural networks or dynamical systems with deviating arguments are obtained. For instances, the differential dynamical system with deviating argument is discussed in detail in [18]. Wu and Zeng investigate the existence and uniqueness of fuzzy neurodynamic system with deviating argument in [19]. Integral manifold and almost periodic solution of a class of differential equation with deviating argument are discussed in [20, 21], respectively. Akhmet et al. explored the stability of several type neural networks with deviating arguments in [22, 23]. In [24], Li investigates the ES of stochastic CNN with deviating argument by using Lyapunov functional. In [25], the stability of semi-linear stochastic differential equation with deviating argument is discussed. We can find that the majority of the researches included in the aforementioned studies, however, focus on the stability of stochastic dynamical systems or neural networks with deviating arguments and no bidirectional associative memory. Few studies have looked at the robustness of stability.

Robustness is the ability of a control system to maintain a specific level of performance when certain parameters are perturbed, and it is of great significance for the design and application of system. As for the analysis of RoS, there are lots of interesting outcomes, such as, Si et al. further studied RoS of dynamical systems with deviating arguments in [26–28]; Fang et al. analyzed the RoS of fuzzy CNN with deviating argument in [29]. In [30], RoS of recurrent neural network is investigated. But, few publications have explored the robustness of stability for BAMCNN with deviating argument (BAMDA). This inspires us to finish this paper. As far as we know, the literature has never addressed the issue of RoS of BAMDA. The following are the works and contributions in this article:(i)This research investigates the necessary circumstances for the existence and uniqueness of BAMDA solutions by constructing equivalent integral equations.(ii)The RoS of BAMDA and stochastic BAMDA (SBAMDA) are investigated by using Gronwall inequality and other inequality techniques, furthermore, the upper bounds of the interval of deviating argument and the intensity of noise before losing its stability are calculated. Furthermore, the mutual constraint between the two disturbances is highlighted.(iii)With the use of the Gronwall–Bellman lemma and inequality techniques, the academic framework for developing BAMDA that fulfills performance standards is provided.

Finally, we list the organizations of this paper. Section 2 introduces the model as well as the lemmas we utilized to get the key outcomes. In Section 3, we will investigate the robustness of BAMDA we proposed. And not only do we make a research on the robustness of SBAMDA but also we derive the maximum of noise intensity and the max extent of the deviating intervals in Section 4. Besides, several numerical instances are given in Section 5 to demonstrate the theoretical values.

Notations: Denote . , , and represent the space of real number, positive real number, and -dimensional vectors, respectively. denotes the Euclidean norm of a real vector. Complete filtered probability space embraces all -null sets, where filtration is right continuous and satisfies the usual conditions. Scalar Brownian movement is defined at . represents the mathematical expectation operator.

2. Introduction

The following system is under consideration in this part, it is formed by BAMCNN with deviating arguments:where , , and , represents the th cell and th cell of time , respectively; positive numbers and represent the rates of the th and th cell adjust to their electricpotential to the static state when loss connections with other cells and inputs from the exterior at time . and are arbitrary numbers, and they represent the connection strength between the th and th cell at time , respectively; and are contrary to the definition of and , which are the connection strengths between the th and th cell at time , respectively. Besides, and are nonlinear activation functions of th and th neuron at time , respectively, and they satisfy and . is a deviating function with , when , where and sequences and satisfy that , , and , if .

Furthermore, from (1), we can get the undisturbed system as follows:

Obviously, system (2) has equilibrium point (0, 0). Because of the arguments may change their deviation characteristic during its movement, thus, the system is delayed or advanced. Consider the interval , , then, , thus, when satisfies , that is, , then is an advanced argument. As the same time, if , that is, , then is an delayed argument. Therefore, we call it mixed type.

We use the following notations for the purpose of simplicity:

The following are the presumptions we will require for this paper:(i)A1: There are two positive constants and such that and this assumption implies that .(ii)A2: There is a satisfies ;(iii)A3: ;(iv)A4: ;

where

Remark 1. If assumptions A3 and A4 hold, then and , and then we can derive that and .
In this paper, we presume that the solutions of the model (1) we proposed above are continuous. Generically, it is discontinuous for the right-side of (1) at the moment , , because of the discontinuity of deviating function at the moment. Thus, based on our assumptions above, the solutions of the equations are viewed as continuously differentiable at the interval , . That means (1) is satisfied by , on , where denotes the solution of equation (1). Besides, and exists one side derivative at and which denotes that , , and .

Lemma 1. Let assumptions A1 to A4 hold, then, for any initial values , there exists a unique solution such that , .

Proof. Existence: For all , we assume that , and denote and for simplicity.
Firstly, for , , thenWe denote 0-norm as , , and we let , , simplicity. And we transform (1) to the following equation:Then, we construct the following sequences and , where , such thatFurthermore,Similarly, we also haveThen,Let , then we can getMoreover,Thus, is the solution of (7) on . Then condition A1 implies that and can be continuous on .
Using the same method, we can continue and from to , and then to . Therefore, we complete this proof of existences by mathematical induction.
Uniqueness: For simplicity, we denote , , , , and we assume and are solutions of (1), where , , , and are initial values and .
From (1), thenFrom (14), thenLet , , and , then (15) can be recorded asApplied Gronwall's inequality, we can getand when , we havethenOn the hand, if and , similarly we can getWe can obtain (21) as followsCombine (19)–(21), we can getObviously, we can see that A5 contradicts (22), and then the uniqueness of solutions of system we proposed is proved for . And it is evident that the extension of the uniqueness of solutions on is available.

3. Deviating Argument Impacts on Stability

Let us introduce the definition of globally exponentially stable (GES) first.

Definition 1. The states of (2) are GES, if for all and , there are , and such thatwhere is the initial value of system (2).
Another assumption must be met before we can proceed to our major results.(i)A5 .To reach our primary results, we first present the following lemma.

Lemma 2. Let A1 to A4 hold, then there exists a such thatwhere and are the solutions of equation (1), and .

Proof. Without losing generality, suppose that , then .
From (1), thenFrom (25),Similarly,ThenBy applying Gronwall–Bellman LemmaSimilarly,therefore,Substituting (29) and (30)where .

Theorem 1. Let assumptions A1–A5 hold, then, BAMNN (2) is GES, when , where

Proof. For simplicity, we denote and as and . From BAMNN (1) and (2),Then,In what follows,Then,From (24) and Definition 1,Using the Gronwall–Bellman Lemma, for ,then for , we havewhereFrom , it is obvious that and , besides, for parameter , is increasing, thus, undoubtedly, we can find a unique to satisfy .
From , we can easily obtain which is also strictly increasing, thus a unique can be fonded such that , then and , so there must be a unique such that , hence, when , we can obtain .
So when , setting , we obtainFrom the uniqueness of the solutions we have proved before, we getwhere . Thus, by (43) and (44) we can be obtained for ,Thus, for all and can be found to satisfy , andClearly, (46) is also hold for , then, the perturbed system (1) is GES.

4. Noise and Deviating Arguments Impacts on Stability

We will analyze the stability of SBAMDA in this part, the system is expressed as follows:where , , , , , , , , , and are same which is defined in (1). is the intensity of disturbances. Based on Section 2, (47) has an unique state for any initial value .

Then, we give the definition of MSES (MSES) of system (47).

Definition 2. System (32) is said to be MSES if there exists and such thatwhere and are the state and initial value of the SBAMDA (47), respectively.

Definition 3. System (32) is called almost GES, if for any , , , the Lyapunov exponent .

Remark 2. It is obvious that, from the definitions above, the mean square GES follows from the almost sure GES, but the opposite is not true. On the other hand, if A1 holds means that the MSES implies almost sure exponential stable (ASES).
Therefore, we give some assumptions we need in this part as follows:(i)A6(ii)A7whereIn the next lemma, we will explore the relationships of state of system (47) and deviating function .

Lemma 3. Let assumption A1–A6 hold, then exists such thatwhere

Proof. Since , thus, there must be a positive constant , satisfied , then , from (32),Furthermore,Similarly,Then,Applying the Gronwall–Bellman Lemma,At the same time,Similarly,Then, from (58) we can havewhere .
Then combine with A6, we haveTherefore, (63) holds on the interval , since the randommicities of and , thus, , , (44) also holds. Therefore, the proof is completed.
In what follows, the implications of both deviating arguments and stochastic disturbances on the stability of BAMNN (2) is examined.

Theorem 2. Let A1 to A7 hold and BAMNN (2) is GES, then, SBAMDA (32) is MSES and also ASES if , , , where and are the unique positive solutions of the following two transcendental equations, respectively,where

Proof. From (2), (24), (25), and (47),Furthermore, we can obtainThenApplying Gronwall–Bellman Lemma, for , we can get thatThus, for ,whereTherefore, we haveand we can easily get (72) is strictly increase for .
From the assumption A7, we can easily obtain thatOn the other hand, we can easily get , thus, suppose that . So, when , , and . Besides, we denote , we can easily get is increasing for . Then, from A6, we know that , so exists a such that , so, when , holds.
And because is strictly increasing for , so exists , holds, when , .
Setting , thusBy Theorem 1, the solution of (47) is unique, thus, there is a , such thatholds for .
Thus, for any , there is a such that ,Clearly, we can easily get (76) is also hold for . Thus, system (47) is satisfies Definition 2. According to Remark 2, (47) is also ASES.

Remark 3. Table 1 is a brief comparison between this paper and existing literature. The factors we compare are RoS, BAM, deviating argument (DA), the existence and uniqueness of solutions (EU), ES, AS.

5. Examples

We provide a few instances in this section to demonstrate the viability of the results in the earlier sections.

Example 1. Consider the following BAMDA, let , the parameters and the model of our example are as follows:where and .
From the parameters, we have , , , , , , , , and .
Moreover, we haveBesides, according to the comparison theorem, we know that the system without deviating argumentis GES with .
Let , , then we can get from the equation as follows:Therefore, when , according to Theorem 1, (47) is GES.
Figure 1 depicts the states of system (77) with , in different initial values. Since the length of interval of deviating argument we take in system (77) is less than 0.1393, thus, BAMDA (77) is GES.

Example 2. Consider the following SBAMDA:where the deviating argument , when , .
By computing the parameters, we can obtain , , , , , , , , , and . Besides, we let , by the comparison theorem, we can obtain that the origin BAMCNN of (81).is GES with , .
Moreover, we haveSubstituting the parameters into (72), we haveLet , thus, we have . Note that , that is , then substituting the other parameters into , then we can getwhereThus, we can obtain , recalling that , and therefore, .
Hence, we can easily get for given stable system, when , , the perturbed system will also be stable.
The states in Figure 2 is the states under the condition of system (81) with , , and . It shows that if the and are both lower than we derived in this paper, then (81) will be exponential stable.
In Figure 3, , , thus, the states in Figure 3 is not exponentially stable. And Figure 4 shows the states of (81), when and is bigger than we derived in this paper, we can easily find that the states in Figure 4 has lost its origin stability.
In Figure 5, we choose and , they are both bigger than the results we derived in this paper, hence, the states of SBAMDA (81) is not GES.

6. Conclusion

The robustness of the stability of BAMCNN with deviating argument and noise is analyzed. The upper bounds of noise and the deviating intervals must be limited in order to retain the original stability. We may estimate these upper bounds that we derived by resolving transcendental equations. Our findings provide a theoretical underpinning for BAMCNN designs and implementations. Future study may focus on enhancing the upper limits and considering employing classical approaches to optimize the computation process, such as the LMI method and the Lyapunov function method. Furthermore, more sophisticated structural disturbances, such as Markov jump, impulses, state-dependent delays, and so on, can be taken into account.

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 Natural Science Foundation of China under Grant nos. 62072164 and 12074111.

References

L. O. Chua and L. Yang, “Cellular neural networks: theory,” IEEE Transactions on Circuits and Systems, vol. 35, no. 10, pp. 1257–1272, 1988.
View at: Publisher Site | Google Scholar
J. Cao and D. Zhou, “Stability analysis of delayed cellular neural networks,” Neural Networks, vol. 11, no. 9, pp. 1601–1605, 1998.
View at: Publisher Site | Google Scholar
A. Chen, J. Cao, and L. Huang, “Global robust stability of interval cellular neural networks with time-varying delays,” Chaos, Solitons & Fractals, vol. 23, no. 3, pp. 787–799, 2005.
View at: Publisher Site | Google Scholar
B. Kosko, “Adaptive bidirectional associative memories,” Applied Optics, vol. 26, no. 23, pp. 4947–4960, 1987.
View at: Publisher Site | Google Scholar
C. Li, X. Yu, T. Huang, G. Chen, and X. He, “A generalized hopfield network for nonsmooth constrained convex optimization: lie derivative approach,” IEEE Transactions on Neural Networks and Learning Systems, vol. 27, no. 2, pp. 308–321, 2016.
View at: Publisher Site | Google Scholar
N. Maglaveras, T. Stamkopoulos, C. Pappas, and M. Strintzis, “Ecg processing techniques based on neural networks and bidirectional associative memories,” Journal of Medical Engineering & Technology, vol. 22, no. 3, pp. 106–111, 1998.
View at: Publisher Site | Google Scholar
H. Xiang and J. Cao, “Exponential stability of periodic solution to cohen-grossberg-type bam networks with time-varying delays,” Neurocomputing, vol. 72, no. 7-9, pp. 1702–1711, 2009.
View at: Publisher Site | Google Scholar
C. Sowmiya, R. Raja, J. Cao et al., “Global exponential stability of antiperiodic solutions for impulsive discrete-time Markovian jumping stochastic bam neural networks with additive time-varying delays and leakage delay,” International Journal of Adaptive Control and Signal Processing, vol. 32, no. 6, pp. 908–936, 2018.
View at: Publisher Site | Google Scholar
C. Sowmiya, R. Raja, J. Cao, G. Rajchakit, and A. Alsaedi, “A delay-dependent asymptotic stability criteria for uncertain bam neural networks with leakage and discrete time-varying delays: a novel summation inequality,” Asian Journal of Control, vol. 22, no. 5, pp. 1880–1891, 2020.
View at: Publisher Site | Google Scholar
X. Liu, N. Jiang, J. Cao, S. Wang, and Z. Wang, “Finite-time stochastic stabilization for bam neural networks with uncertainties,” Journal of the Franklin Institute, vol. 350, no. 8, pp. 2109–2123, 2013.
View at: Publisher Site | Google Scholar
A. P. Chen, J. Cao, and L. H. Huang, “Exponential stability of bam neural networks with transmission delays,” Neurocomputing, vol. 57, pp. 435–454, 2004.
View at: Publisher Site | Google Scholar
H. Bao and J. Cao, “Exponential stability for stochastic bam networks with discrete and distributed delays,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6188–6199, 2012.
View at: Publisher Site | Google Scholar
P. Balasubramaniam and C. Vidhya, “Global asymptotic stability of stochastic bam neural networks with distributed delays and reaction-diffusion terms,” Journal of Computational and Applied Mathematics, vol. 234, no. 12, pp. 3458–3466, 2010.
View at: Publisher Site | Google Scholar
J. D. Cao, “Exponential stability of delayed bi-directional associative memory networks,” Applied Mathematics and Computation, vol. 135, pp. 105–112, 2003.
View at: Google Scholar
Q. Zhu, C. Huang, and X. Yang, “Exponential stability for stochastic jumping bam neural networks with time-varying and distributed delays,” Nonlinear Analysis: Hybrid Systems, vol. 5, no. 1, pp. 52–77, 2011.
View at: Publisher Site | Google Scholar
Q. Zhu and J. Cao, “Stability analysis of Markovian jump stochastic bam neural networks with impulse control and mixed time delays,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 3, pp. 467–479, 2012.
View at: Publisher Site | Google Scholar
K. S. Chiu and M. Pinto, “Periodic solutions of differential equations with a general piecewise constant argument and applications,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 46, pp. 1–19, 08 2010.
View at: Publisher Site | Google Scholar
S. M. Shah and J. Wiener, “Advanced differential equations with piecewise constant argument deviations,” International Journal of Mathematics and Mathematical Sciences, vol. 6, no. 4, pp. 671–703, 1983.
View at: Publisher Site | Google Scholar
A. Wu and Z. Zeng, “Output convergence of fuzzy neurodynamic system with piecewise constant argument of generalized type and time-varying input,” IEEE Transactions on Systems, Man, and Cybernetics: Systems, vol. 46, no. 12, pp. 1689–1702, 2016.
View at: Publisher Site | Google Scholar
M. Akhmet, “Almost periodic solutions of differential equations with piecewise constant argument of generalized type,” Nonlinear Analysis: Hybrid Systems, vol. 2, no. 2, pp. 456–467, 2008.
View at: Publisher Site | Google Scholar
M. Akhmet, “Integral manifolds of differential equations with piecewise constant argument of generalized type,” Nonlinear Analysis: Theory, Methods & Applications, vol. 66, no. 2, pp. 367–383, 2007.
View at: Publisher Site | Google Scholar
M. Akhmet and E. Yılmaz, “Impulsive hopfield-type neural network system with piecewise constant argument,” Nonlinear Analysis: Real World Applications, vol. 11, no. 4, pp. 2584–2593, 2010.
View at: Publisher Site | Google Scholar
M. Akhmet, D. Aruğaslan, and E. Yılmaz, “Stability in cellular neural networks with a piecewise constant argument,” Journal of Computational and Applied Mathematics, vol. 233, no. 9, pp. 2365–2373, 2010.
View at: Publisher Site | Google Scholar
X. Li, “Existence and exponential stability of solutions for stochastic cellular neural networks with piecewise constant argument,” Journal of Applied Mathematics, vol. 2014, Article ID 145061, 11 pages, 2014.
View at: Publisher Site | Google Scholar
Z. Ling, “Stability analysis for semi-linear stochastic differential equation with piecewise continuous arguments,” Journal of Physics: Conference Series, vol. 2012, Article ID 012088, 2012.
View at: Publisher Site | Google Scholar
W. X. Si, T. Xie, and B. W. Li, “Robustness analysis of exponential stability of neutral-type nonlinear systems with multi-interference,” IEEE Access, vol. 9, pp. 116015–116032, 2021.
View at: Publisher Site | Google Scholar
W. Si, T. Xie, B. Li, and S. Wen, “Exploration on robustness of exponentially global stability of recurrent neural networks with neutral terms and generalized piecewise constant arguments,” Discrete Dynamics in Nature and Society, vol. 2021, Article ID 9941881, 13 pages, 2021.
View at: Publisher Site | Google Scholar
W. Si, T. Xie, B. Li, and Y. Song, “Further results on exponentially robust stability of uncertain connection weights of neutral-type recurrent neural networks,” Complexity, vol. 2021, Article ID 6941701, 2115 pages, 2021.
View at: Publisher Site | Google Scholar
F. Wenxiang, X. Tao, and L. Biwen, “Robustness analysis of fuzzy cellular neural network with deviating argument and stochastic disturbances,” IEEE Access, vol. 11, pp. 3717–3728, 2023.
View at: Publisher Site | Google Scholar
Y. Shen and J. Wang, “Robustness analysis of global exponential stability of recurrent neural networks in the presence of time delays and random disturbances,” IEEE Transactions on Neural Networks and Learning Systems, vol. 23, no. 1, pp. 87–96, 2012.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2023 Wenxiang Fang 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.

PDF Download Citation

Download other formats

Order printed copies

Views

202

Downloads

242

Citations