Multistability of delayed fractional-order competitive neural networks

doi:10.1016/j.neunet.2021.03.036

Neural Networks

Volume 140, August 2021, Pages 325-335

https://doi.org/10.1016/j.neunet.2021.03.036 Get rights and content

Abstract

This paper is concerned with the multistability of fractional-order competitive neural networks (FCNNs) with time-varying delays. Based on the division of state space, the equilibrium points (EPs) of FCNNs are given. Several sufficient conditions and criteria are proposed to ascertain the multiple $O (t^{- α})$ -stability of delayed FCNNs. The $O (t^{- α})$ -stability is the extension of Mittag-Leffler stability of fractional-order neural networks, which contains monostability and multistability. Moreover, the attraction basins of the stable EPs of FCNNs are estimated, which shows the attraction basins of the stable EPs can be larger than the divided subsets. These conditions and criteria supplement and improve the previous results. Finally, the results are illustrated by the simulation examples.

Introduction

In the last few decades, the study of neural networks has gradually attracted much attention due to the potential applications in pattern recognition, associative memory, parallel computation, classification, and optimization (Chua and Yang, 1988, Luo et al., 2020, Wang et al., 2018). In the applications of neural networks, neural networks are required to have the stable dynamics in Chen and Liu, 2017, Forti and Tesi, 2004, Srivastava et al., 2018, Tyagi and Abbas, 2017 and Zhang and Zeng, 2018, Zhang and Zeng, 2020, Zhang and Zeng, 2021a. In recent decades, the research of fractional-order calculus has been extended to the field of science and engineering application. Many related achievements have greatly promoted the research progress of fractional calculus in Samko, Kilbas, and Marichev (1993). Due to the weak heredity, weak memory and inaccurate modeling of neural networks, fractional calculus can effectively make up for the deficiencies of neural networks in these aspects. With the introduction of fractional calculus into neural networks, it can be predicted that the research and application of fractional-order neural networks will have a new leap.

The multistability of neural networks is the complex dynamic behavior, and it is related to the local stable regions. In particular, many valuable results have been obtained for the stability of the EPs and periodic orbits of neural networks. For example, based on the partition of state space, several sufficient conditions of the coexistence and stability of EPs were derived by the local characteristics of activation functions of neural networks (Cheng et al., 2015, Zeng and Zheng, 2013). The multiple $ψ$ -type stability and robust stability of neural networks (Zhang and Zeng, 2019a, Zhang and Zeng, 2019b, Zhang and Zeng, 2021b) were given by improving the relative convergence rate. Based on some theoretical research and practice, the multiple Mittag–Leffler stability of fractional-order neural networks was proved under the conditions in Liu, Zeng, and Wang (2017) and Wan and Wu (2018). Moreover, the attraction basins of the stable EPs of delayed neural networks (Cheng et al., 2007, Liu, Zeng, and Wang, 2018) were extended on the basis of the divided regions. Furthermore, many other valuable works are found in Guo et al., 2019, Hu et al., 2019, Song and Chen, 2018 and Zhang and Zeng, 2016, Zhang and Zeng, 2021c and the references cited therein.

Because of the power-law phenomenon and memory characteristics of fractional calculus, the study of fractional-order systems (Chen et al., 2019, Li et al., 2009, Liang et al., 2015, Liu et al., 2019, Tyagi et al., 2016, Tyagi et al., 2017a, Tyagi et al., 2017b, Wan and Jian, 2019, Yang et al., 2020) has been widely concerned. For example, the state variables of integer-order systems may decay as the form of the exponential functions, while the actual detected results may be the power functions, which is called power-law phenomenon. Due to the imprecise modeling, the phenomenon is not understood in integer-order systems. The phenomenon is easy to understand under the framework of fractional-order systems. Fractional calculus is a good mathematical tool for people to better understand the complex dynamics of systems. Therefore, the introduction and research of fractional-order neural networks can open up a new way for the research and application of neural networks.

In the research area of the dynamic behaviors of competitive neural networks, several theoretical results have been derived in some literatures (see Lu and Amari, 2006, Meyer-Base and Thummler, 2008, Nie et al., 2019, Pratap et al., 2019, Yang et al., 2010, Yang et al., 2017). The synchronization of competitive neural networks with the switched connection weight was discussed by analytical methods in Gong, Yang, Guo, and Huang (2019) and Pratap, Raja, Cao, Rajchakit, and Alsaadi (2018). The robust stability and asymptotic stability of competitive neural networks with time-scale (Liu, Yang, and Zhou, 2018, Meyer-Baese et al., 2007) were proved. Moreover, based on the partition of state space, the multistability and multiperiodicity of neural networks (Nie and Huang, 2012, Wang and Luo, 2015, Xie and Zheng, 2015) with respect to EPs and periodic orbits were improved by comparing similar results. Furthermore, the multistability of integer-order competitive neural networks was generalized to the multiple Mittag–Leffler stability of FCNNs with Gaussian activation functions in Liu, Nie, Liang, and Cao (2018). It should be pointed out that these results mainly focus on the stability analysis of integer-order competitive neural networks and the multiple Mittag–Leffler stability of FCNNs without delays. However, little attention has been paid to the multistability of delayed FCNNs.

Generally speaking, as pointed out in Diethelm (2010), the asymptotic expressions of Mittag–Leffler function could approximate inverse-power-law at infinity. Compared with the Mittag–Leffler stability of fractional-order system, $t^{- α}$ -stability can intuitively describe the relative convergence rate of system. Previously, the $α$ -stability and $α$ -synchronization of fractional-order neural networks (Chen and Chen, 2016, Li et al., 2013, Yu et al., 2012) were explored and analyzed. On this basis, the sufficient conditions related to Lagrange $α$ -exponential stability were derived in Jian and Wan (2017). In addition, the $O (t^{- α})$ -stability and $O (t^{- α})$ -synchronization of fractional-order neural networks were investigated and discussed in Chen, Chen, and Zeng (2018) and Rakkiyappan, Sivaranjani, Velmurugan, and Cao (2016). It is worth noting that these results mainly focus on $t^{- α}$ -synchronization and monostability of fractional-order neural networks, and there are few researches on the multistability and attraction basin of fractional-order neural networks.

Bearing the above discussion in mind, $O (t^{- α})$ -stability is introduced, and whether FCNNs have $O (t^{- α})$ -stability. On the one hand, fractional-order neurons can be simulated and approximated to integer-order neurons, and competitive neural networks composed of fractional-order neurons are helpful to study and explore the internal mechanism of neural networks. On the other hand, the multiple $O (t^{- α})$ -stability is an extension of the multiple Mittag–Leffler stability of fractional-order neural networks, which further enriches the dynamic behaviors of fractional-order neural networks. The fractional-order model used in this paper is a more general model including fractional-order Hopfield neural networks with or without delays. By comparing several related results, the obtained conditions and criteria further expand the previous results.

The rest of the paper is organized as follows. In Section 2, the preliminaries are given so that will be used later. In Section 3, several sufficient conditions are derived for the coexistence and $O (t^{- α})$ -stability of the EPs of FCNNs with time-varying delays. Meanwhile, the attraction basins of $O (t^{- α})$ -stable EPs are estimated. Two examples are given to demonstrate the effectiveness of the obtained results in Section 4. Finally, the conclusion is drawn in Section 5.

Section snippets

Preliminaries

Definition 1

${‖ {(x^{T}, y^{T})}^{T} ‖}_{{ξ, \bar{ξ}, \infty}} ≔ {max}_{i \in N} {| ξ_{i}^{- 1} x_{i} |, | {\bar{ξ}}_{i}^{- 1} y_{i} |}$ , where $ξ_{i} > 0$ , ${\bar{ξ}}_{i} > 0$ , $ξ = {(ξ_{1}, ξ_{2}, \dots, ξ_{n})}^{T}$ , $\bar{ξ} = {({\bar{ξ}}_{1}, {\bar{ξ}}_{2}, \dots, {\bar{ξ}}_{n})}^{T}$ , $x = {(x_{1}, x_{2}, \dots, x_{n})}^{T}$ , $y = {(y_{1}, y_{2}, \dots, y_{n})}^{T}$ and $i \in N ≔ {1, 2, \dots, n}$ .

Definition 2

The Caputo fractional-order derivative with order $α$ for function $f : [t_{0}, + \infty) \to R$ is defined as $_{t_{0}}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (m - α)} \int_{t_{0}}^{t} \frac{f^{(n)} (s)}{{(t - s)}^{α - n + 1}} d s,$ where $t \geq t_{0}$ , $n - 1 < α < n$ and $n$ is a positive integer. In particular, when $0 < α < 1$ , $_{t_{0}}^{C} D_{t}^{α} f (t) = \frac{1}{Γ (1 - α)} \int_{t_{0}}^{t} \frac{f^{'} (s)}{{(t - s)}^{α}} d s .$ where Caputo derivative $_{0}^{C} D_{t}^{α} f (t)$ with $0 < α < 1$ is written as $D^{α} f (t)$ .

In this paper, consider a general

Coexistence of the EPs of FCNNs

Lemma 1

If functions $F_{i I} (\cdot)$ and $F_{i S} (\cdot)$ for the sequence of points ${q_{i}^{(k)}}_{k = 0}^{K_{i}}$ , ${p_{i}^{(k)}}_{k = 0}^{K_{i}}$ and $i \in N$ satisfy $F_{i I} (q_{i}^{(k)}) > 0, F_{i S} (p_{i}^{(k)}) < 0,$ then each set in $Ω_{1}$ has at least the EP of FCNN (1).

Proof

For $x = {(x_{1}, x_{2}, \dots, x_{n})}^{T} \in R^{n}$ and $i \in N$ , let $\{\begin{matrix} F_{i}^{(1)} (r) = - d_{i} r + (a_{i i} + b_{i i} + c_{i}^{- 1} a_{i} b_{i}) f_{i} (r) \\ + \sum_{j = 1, j \neq i}^{n} (a_{i j} + b_{i j}) f_{j} (x_{j}) + I_{i}, \\ F_{i}^{(2)} (ρ, r) = - c_{i} ρ + b_{i} f_{i} (r) . \end{matrix}$ From (5), for $k = 0, 1, \dots, K_{i} + 1$ , $p_{i}^{(- 1)} = - \infty$ and $q_{i}^{(K_{i} + 1)} = + \infty$ , $F_{i}^{(1)} (q_{i}^{(k)}) \geq F_{i I} (q_{i}^{(k)}) > 0, F_{i}^{(1)} (p_{i}^{(k - 1)}) \leq F_{i S} (p_{i}^{(k - 1)}) < 0 .$ Without loss of generality, since the partial solution $r$ is separable and the

Numerical example

This section exemplifies our theoretical results with two examples.

Example 1

Consider the following FCNN (1) $\{\begin{matrix} D^{α} x_{1} (t) = & - x_{1} (t) + 2 f_{1} (x_{1} (t)) + 0.01 f_{1} (x_{1} (t - τ (t))) \\ - 0.1 f_{2} (x_{2} (t)) + 0.1 y_{1} (t) - 1.88 \\ - 0.02 f_{2} (x_{2} (t - τ (t))) \\ D^{α} x_{2} (t) = & - 1.5 x_{2} (t) + f_{2} (x_{2} (t)) + 0.03 f_{2} (x_{2} (t - τ (t))) \\ + 0.2 f_{1} (x_{1} (t)) - 0.2 y_{2} (t) + 1.2 \\ - 0.02 f_{1} (x_{1} (t - τ (t))) \\ D^{α} y_{1} (t) = & - y_{1} (t) + f_{1} (x_{1} (t)), \\ D^{α} y_{2} (t) = & - y_{2} (t) + f_{2} (x_{2} (t)) . \end{matrix}$ where $α = 0.98$ , $τ (t) = \frac{t}{1 + t}$ and $f_{i} (r) = f (r) = exp {- r^{2}}$ for $i = 1, 2$ . There exist numbers $M_{i} = 1$ and $m_{i} = 0$ for continuous activation functions, which implies ${\bar{m}}_{i} = 0$ and ${\bar{M}}_{i} = 1$ for $i = 1, 2$

Conclusion

In this paper, the multiple $O (t^{- α})$ -stability of delayed FCNNs is discussed. The criteria on the $O (t^{- α})$ -stability of delayed FCNNs are derived by the weighted norm, which avoids the construction of Lyapunov function. Furthermore, the attraction basins of the $O (t^{- α})$ -stable EPs of FCNNs are estimated, and these attraction basins could be larger than the divided subsets. The $O (t^{- α})$ -stability is the extension of Mittag–Leffler stability of fractional-order neural networks, and it contains

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

The work was supported by the Natural Science Foundation of China under Grants 61936004 and 61673188, China Postdoctoral Science Foundation under Grant 2019M652645, the Innovation Group Project of the National Natural Science Foundation of China under Grant 61821003, the Foundation for Innovative Research Groups of Hubei Province of China under Grant 2017CFA005 and the 111 Project on Computational Intelligence and Intelligent Control, China under Grant B18024.

Fanghai Zhang received the B.S. degree in mathematics and applied mathematics from Fuyang Normal College, Fuyang, China, in 2012, and the Ph.D. degree in control science and engineering from Huazhong University of Science and Technology, Wuhan, China, in 2018. He is currently a Post-Doctoral Researcher in control science and engineering with the School of Automation and the School of Computer Science and Technology, Huazhong University of Science and Technology. His current research interests

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Cited by (0)

Tingwen Huang is a Professor at Texas A&M University at Qatar. He received his B.S. degree from Southwest Normal University (now Southwest University), China, 1990, his M.S. degree from Sichuan University, China, 1993, and his Ph.D. degree from Texas A&M University, College Station, Texas, 2002. After graduated from Texas A&M University, he worked as a Visiting Assistant Professor there. Then he joined Texas A&M University at Qatar (TAMUQ) as an Assistant Professor in August 2003, then he was promoted to Professor in 2013. Dr. Huang’s focus areas for research interests include neural networks, computational intelligence, chaotic dynamical systems, omplex networks, optimization and control.

Qiujie Wu received M.S degree in the School of Automation, Huazhong University of Science and Technology, Wuhan, Hubei, China, in 2016, and Ph.D. degree in School of Artificial Intelligence and Automation from Huazhong University of Science and Technology, Wuhan, China, in 2020. She is currently a teacher with the School of Internet, Anhui University, Hefei, China. Her current research interests include chaotic system and its application to secure communication.

Zhigang Zeng received the Ph.D. degree in systems analysis and integration from Huazhong University of Science and Technology, Wuhan, China, in 2003. He is currently a Professor with the School of Automation, Huazhong University of Science and Technology, Wuhan, China, and also with the Key Laboratory of Image Processing and Intelligent Control of the Education Ministry of China, Wuhan, China. He has been an Associate Editor of the IEEE Transactions on Neural Networks (2010-2011), IEEE Transactions on Cybernetics (since 2014), IEEE Transactions on Fuzzy Systems (since 2016), and a member of the Editorial Board of Neural Networks (since 2012), Cognitive Computation (since 2010), Applied Soft Computing (since 2013). He has published over 100 international journal papers. His current research interests include theory of functional differential equations and differential equations with discontinuous righthand sides, and their applications to dynamics of neural networks, memristive systems, and control systems.

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Multistability of delayed fractional-order competitive neural networks

Abstract

Introduction

Section snippets

Preliminaries

Coexistence of the EPs of FCNNs

Numerical example

Conclusion

Declaration of Competing Interest

Acknowledgments

Neural Networks

Neural Networks

Physica D. Nonlinear Phenomena

Neural Networks

Automatica

Neurocomputing

Neural Networks

Neurocomputing

Neurocomputing

Applied Mathematics and Computation

Neurocomputing

Journal of the Franklin Institute. Engineering and Applied Mathematics

Neural Networks

Neurocomputing

Neurocomputing

Neural Networks

Neurocomputing

Neural Networks

Global asymptotic stability and adaptive ultimate Mittag-Leffler synchronization for a fractional-order complex-valued memristive neural networks with delays

IEEE Transactions on Systems, Man & Cybernetics, Part A (Systems & Humans)

μ-stability of nonlinear positive systems with unbounded time-varying delays

IEEE Transactions on Neural Networks and Learning Systems

Multistability for delayed neural networks via sequential contracting

IEEE Transactions on Neural Networks and Learning Systems

Cellular neural networks: Applications

IEEE Transactions on Circuits and Systems

The analysis of fractional differential equations

Absolute stability of analytic neural networks: An approach based on finite trajectory length

IEEE Transactions on Circuits and Systems I

Global exponential synchronization of memristive competitive neural networks with time-varying delay via nonlinear control

Neural Processing Letters

Multistability of switched neural networks with piecewise linear activation functions under state-dependent switching

IEEE Transactions on Neural Networks and Learning Systems

Multistability of delayed hybrid impulsive neural networks with application to associative memories

IEEE Transactions on Neural Networks and Learning Systems

$μ$ -stability of nonlinear positive systems with unbounded time-varying delays