1
|
Zhang F, Huang T, Wu A, Zeng Z. Mittag-Leffler stability and application of delayed fractional-order competitive neural networks. Neural Netw 2024; 179:106501. [PMID: 38986190 DOI: 10.1016/j.neunet.2024.106501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/23/2024] [Revised: 06/04/2024] [Accepted: 06/28/2024] [Indexed: 07/12/2024]
Abstract
In the article, the Mittag-Leffler stability and application of delayed fractional-order competitive neural networks (FOCNNs) are developed. By virtue of the operator pair, the conditions of the coexistence of equilibrium points (EPs) are discussed and analyzed for delayed FOCNNs, in which the derived conditions of coexistence improve the existing results. In particular, these conditions are simplified in FOCNNs with stepped activations. Furthermore, the Mittag-Leffler stability of delayed FOCNNs is established by using the principle of comparison, which enriches the methodologies of fractional-order neural networks. The results on the obtained stability can be used to design the horizontal line detection of images, which improves the practicability of image detection results. Two simulations are displayed to validate the superiority of the obtained results.
Collapse
Affiliation(s)
- Fanghai Zhang
- School of Electrical Engineering and Automation, Hefei University of Technology, Hefei 230009, China.
| | - Tingwen Huang
- Department of Science Program, Texas A&M University at Qatar, Doha, Qatar.
| | - Ailong Wu
- College of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China.
| | - Zhigang Zeng
- School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China.
| |
Collapse
|
2
|
Di Marco M, Forti M, Pancioni L, Tesi A. On convergence properties of the brain-state-in-a-convex-domain. Neural Netw 2024; 178:106481. [PMID: 38945117 DOI: 10.1016/j.neunet.2024.106481] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/11/2024] [Revised: 05/14/2024] [Accepted: 06/19/2024] [Indexed: 07/02/2024]
Abstract
Convergence in the presence of multiple equilibrium points is one of the most fundamental dynamical properties of a neural network (NN). Goal of the paper is to investigate convergence for the classic Brain-State-in-a-Box (BSB) NN model and some of its relevant generalizations named Brain-State-in-a-Convex-Body (BSCB). In particular, BSCB is a class of discrete-time NNs obtained by projecting a linear system onto a convex body of Rn. The main result in the paper is that the BSCB is convergent when the matrix of the linear system is symmetric and positive semidefinite or, otherwise, it is symmetric and the step size does not exceed a given bound depending only on the minimum eigenvalue of the matrix. This result generalizes previous results in the literature for BSB and BSCB and it gives a solid foundation for the use of BSCB as a content addressable memory (CAM). The result is proved via Lyapunov method and LaSalle's Invariance Principle for discrete-time systems and by using some fundamental inequalities enjoyed by the projection operator onto convex sets as Bourbaki-Cheney-Goldstein inequality.
Collapse
Affiliation(s)
- Mauro Di Marco
- Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy.
| | - Mauro Forti
- Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy.
| | - Luca Pancioni
- Department of Information Engineering and Mathematics, University of Siena, 53100 Siena, Italy.
| | - Alberto Tesi
- Department of Information Engineering, University of Florence, via S. Marta 3 50139 Firenze, Italy.
| |
Collapse
|
3
|
Huang C, Mo S, Cao J. Detections of bifurcation in a fractional-order Cohen-Grossberg neural network with multiple delays. Cogn Neurodyn 2024; 18:1379-1396. [PMID: 38826673 PMCID: PMC11143155 DOI: 10.1007/s11571-023-09934-2] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/10/2021] [Revised: 12/25/2022] [Accepted: 01/24/2023] [Indexed: 03/06/2023] Open
Abstract
The dynamics of integer-order Cohen-Grossberg neural networks with time delays has lately drawn tremendous attention. It reveals that fractional calculus plays a crucial role on influencing the dynamical behaviors of neural networks (NNs). This paper deals with the problem of the stability and bifurcation of fractional-order Cohen-Grossberg neural networks (FOCGNNs) with two different leakage delay and communication delay. The bifurcation results with regard to leakage delay are firstly gained. Then, communication delay is viewed as a bifurcation parameter to detect the critical values of bifurcations for the addressed FOCGNN, and the communication delay induced-bifurcation conditions are procured. We further discover that fractional orders can enlarge (reduce) stability regions of the addressed FOCGNN. Furthermore, we discover that, for the same system parameters, the convergence time to the equilibrium point of FONN is shorter (longer) than that of integer-order NNs. In this paper, the present methodology to handle the characteristic equation with triple transcendental terms in delayed FOCGNNs is concise, neoteric and flexible in contrast with the prior mechanisms owing to skillfully keeping away from the intricate classified discussions. Eventually, the developed analytic results are nicely showcased by the simulation examples.
Collapse
Affiliation(s)
- Chengdai Huang
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000 China
| | - Shansong Mo
- School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000 China
| | - Jinde Cao
- School of Mathematics, Southeast University, Nanjing, 210096 China
- Yonsei Frontier Lab, Yonsei University, Seoul, 03722 South Korea
| |
Collapse
|
4
|
Stamov T, Stamov G, Stamova I, Gospodinova E. Lyapunov approach to manifolds stability for impulsive Cohen-Grossberg-type conformable neural network models. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:15431-15455. [PMID: 37679186 DOI: 10.3934/mbe.2023689] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 09/09/2023]
Abstract
In this paper, motivated by the advantages of the generalized conformable derivatives, an impulsive conformable Cohen-Grossberg-type neural network model is introduced. The impulses, which can be also considered as a control strategy, are at fixed instants of time. We define the notion of practical stability with respect to manifolds. A Lyapunov-based analysis is conducted, and new criteria are proposed. The case of bidirectional associative memory (BAM) network model is also investigated. Examples are given to demonstrate the effectiveness of the established results.
Collapse
Affiliation(s)
- Trayan Stamov
- Department of Engineering Design, Technical University of Sofia, Sofia 1000, Bulgaria
| | - Gani Stamov
- Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio TX 78249, USA
| | - Ivanka Stamova
- Department of Mathematics, University of Texas at San Antonio, One UTSA Circle, San Antonio TX 78249, USA
| | - Ekaterina Gospodinova
- Department of Computer Sciences, Technical University of Sofia, Sliven 8800, Bulgaria
| |
Collapse
|
5
|
Wu Z, Nie X, Cao B. Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays. Neural Netw 2023; 160:132-147. [PMID: 36640489 DOI: 10.1016/j.neunet.2022.12.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Revised: 11/09/2022] [Accepted: 12/16/2022] [Indexed: 01/05/2023]
Abstract
This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m1+m2=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.
Collapse
Affiliation(s)
- Zhongwen Wu
- School of Mathematics, Southeast University, Nanjing, 211189, China.
| | - Xiaobing Nie
- School of Mathematics, Southeast University, Nanjing, 211189, China.
| | - Boqiang Cao
- School of Mathematics, Southeast University, Nanjing, 211189, China.
| |
Collapse
|
6
|
Kao Y, Cao Y, Chen X. Global Mittag-Leffler synchronization of coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks via sliding mode control. CHAOS (WOODBURY, N.Y.) 2022; 32:113123. [PMID: 36456319 DOI: 10.1063/5.0102787] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/12/2022] [Accepted: 09/19/2022] [Indexed: 06/17/2023]
Abstract
This paper studies the sliding mode control method for coupled delayed fractional reaction-diffusion Cohen-Grossberg neural networks on a directed non-strongly connected topology. A novel fractional integral sliding mode surface and the corresponding control law are designed to realize global Mittag-Leffler synchronization. The sufficient conditions for synchronization and reachability of the sliding mode surface are derived via the hierarchical method and the Lyapunov method. Finally, simulations are provided to verify our theoretical findings.
Collapse
Affiliation(s)
- Yonggui Kao
- Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China
| | - Yue Cao
- Department of Mathematics, Harbin Institute of Technology, Weihai 264209, China
| | - Xiangyong Chen
- School of Automation and Electrical Engineering, Linyi University, Linyi 276005, China
| |
Collapse
|
7
|
Wan L, Liu Z. Multimode function multistability for Cohen-Grossberg neural networks with mixed time delays. ISA TRANSACTIONS 2022; 129:179-192. [PMID: 34991879 DOI: 10.1016/j.isatra.2021.11.046] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/14/2021] [Revised: 11/18/2021] [Accepted: 11/18/2021] [Indexed: 06/14/2023]
Abstract
In this paper, we are concerned with the multimode function multistability for Cohen-Grossberg neural networks (CGNNs) with mixed time delays. It is introduced the multimode function multistability as well as its specific mathematical expression, which is a generalization of multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and asymptotic stability. Also, according to the neural network (NN) model and the maximum and minimum values of activation functions, n pairs of upper and lower boundary functions are obtained. Via the locations of the zeros of the n pairs of upper and lower boundary functions, the state space is divided into ∏i=1n(2Hi+1) parts correspondingly. By virtue of the reduction to absurdity, continuity of function, Brouwer's fixed point theorem and Lyapunov stability theorem, the criteria for multimode function multistability are acquired. Multiple types of multistability, including multiple exponential stability, multiple polynomial stability, multiple logarithmic stability, and multiple asymptotic stability, can be achieved by selecting different types of function P(t). Two numerical examples are offered to substantiate the generality of the obtained criteria over the existing results.
Collapse
Affiliation(s)
- Liguang Wan
- School of Electrical Engineering and Automation, Hubei Normal University, Huangshi 435002, China; School of information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China.
| | - Zhenxing Liu
- School of information Science and Engineering, Wuhan University of Science and Technology, Wuhan 430081, China.
| |
Collapse
|
8
|
Zhang F, Zeng Z. Multistability and Stabilization of Fractional-Order Competitive Neural Networks With Unbounded Time-Varying Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:4515-4526. [PMID: 33630741 DOI: 10.1109/tnnls.2021.3057861] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
This article investigates the multistability and stabilization of fractional-order competitive neural networks (FOCNNs) with unbounded time-varying delays. By utilizing the monotone operator, several sufficient conditions of the coexistence of equilibrium points (EPs) are obtained for FOCNNs with concave-convex activation functions. And then, the multiple μ -stability of delayed FOCNNs is derived by the analytical method. Meanwhile, several comparisons with existing work are shown, which implies that the derived results cover the inverse-power stability and Mittag-Leffler stability as special cases. Moreover, the criteria on the stabilization of FOCNNs with uncertainty are established by designing a controller. Compared with the results of fractional-order neural networks, the obtained results in this article enrich and improve the previous results. Finally, three numerical examples are provided to show the effectiveness of the presented results.
Collapse
|
9
|
Wan L, Liu Z. Multiple O(t-q) stability and instability of time-varying delayed fractional-order Cohen-Grossberg neural networks with Gaussian activation functions. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.05.018] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
|