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Chen J, Xiao M, Wan Y, Huang C, Xu F. Dynamical Bifurcation for a Class of Large-Scale Fractional Delayed Neural Networks With Complex Ring-Hub Structure and Hybrid Coupling. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:2659-2669. [PMID: 34495847 DOI: 10.1109/tnnls.2021.3107330] [Citation(s) in RCA: 4] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/04/2023]
Abstract
Real neural networks are characterized by large-scale and complex topology. However, the current dynamical analysis is limited to low-dimensional models with simplified topology. Therefore, there is still a huge gap between neural network theory and its application. This article proposes a class of large-scale neural networks with a ring-hub structure, where a hub node is connected to n peripheral nodes and these peripheral nodes are linked by a ring. In particular, there exists a hybrid coupling mode in the network topology. The mathematical model of such systems is described by fractional-order delayed differential equations. The aim of this article is to investigate the local stability and Hopf bifurcation of this high-dimensional neural network. First, the Coates flow graph is employed to obtain the characteristic equation of the linearized high-dimensional neural network model, which is a transcendental equation including multiple exponential items. Then, the sufficient conditions ensuring the stability of equilibrium and the existence of Hopf bifurcation are achieved by taking time delay as a bifurcation parameter. Finally, some numerical examples are given to support the theoretical results. It is revealed that the increasing time delay can effectively induce the occurrence of periodic oscillation. Moreover, the fractional order, the self-feedback coefficient, and the number of neurons also have effects on the onset of Hopf bifurcation.
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Bifurcation Mechanism for Fractional-Order Three-Triangle Multi-delayed Neural Networks. Neural Process Lett 2022. [DOI: 10.1007/s11063-022-11130-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/24/2022]
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3
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Mao X, Wang X, Qin H. Stability analysis of quaternion-valued BAM neural networks fractional-order model with impulses and proportional delays. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2022.08.059] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/15/2022]
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4
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Yazgan R, Hajjaji S, Chérif F. Weighted Pseudo Almost-Automorphic Solutions of Quaternion-Valued RNNs With Mixed Delays. Neural Process Lett 2022. [DOI: 10.1007/s11063-022-10890-x] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/18/2022]
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Mawhin J. Degree, quaternions and periodic solutions. PHILOSOPHICAL TRANSACTIONS. SERIES A, MATHEMATICAL, PHYSICAL, AND ENGINEERING SCIENCES 2021; 379:20190378. [PMID: 33390078 DOI: 10.1098/rsta.2019.0378] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Accepted: 04/20/2020] [Indexed: 06/12/2023]
Abstract
The paper computes the Brouwer degree of some classes of homogeneous polynomials defined on quaternions and applies the results, together with a continuation theorem of coincidence degree theory, to the existence and multiplicity of periodic solutions of a class of systems of quaternionic valued ordinary differential equations. This article is part of the theme issue 'Topological degree and fixed point theories in differential and difference equations'.
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Affiliation(s)
- Jean Mawhin
- Institut de Recherche en Mathématique et Physique, Université catholique de Louvain, Louvain-la-Neuve, Belgium
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Gholami Y. Existence and global asymptotic stability criteria for nonlinear neutral-type neural networks involving multiple time delays using a quadratic-integral Lyapunov functional. ADVANCES IN DIFFERENCE EQUATIONS 2021; 2021:112. [PMID: 33619432 PMCID: PMC7888700 DOI: 10.1186/s13662-021-03274-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 12/07/2020] [Accepted: 02/03/2021] [Indexed: 06/12/2023]
Abstract
In this paper we consider a standard class of the neural networks and propose an investigation of the global asymptotic stability of these neural systems. The main aim of this investigation is to define a novel Lyapunov functional having quadratic-integral form and use it to reach a stability criterion for the under study neural networks. Since some fundamental characteristics, such as nonlinearity, including time-delays and neutrality, help us design a more realistic and applicable model of neural systems, we will use all of these factors in our neural dynamical systems. At the end, some numerical simulations are presented to illustrate the obtained stability criterion and show the essential role of the time-delays in appearance of the oscillations and stability in the neural networks.
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Affiliation(s)
- Yousef Gholami
- Department of Applied Mathematics, Sahand University of Technology, P.O. Box: 51335-1996, Tabriz, Iran
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Arenas AJ, González-Parra G, De La Espriella N. Nonlinear dynamics of a new seasonal epidemiological model with age-structure and nonlinear incidence rate. COMPUTATIONAL AND APPLIED MATHEMATICS 2021; 40:46. [PMCID: PMC7877536 DOI: 10.1007/s40314-021-01430-9] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/25/2020] [Revised: 01/18/2021] [Accepted: 01/20/2021] [Indexed: 06/28/2023]
Abstract
In this article, we study the dynamics of a new proposed age-structured population mathematical model driven by a seasonal forcing function that takes into account the variability of the climate. We introduce a generalized force of infection function to study different potential disease outcomes. Using nonlinear analysis tools and differential inequalities theorems, we obtain sufficient conditions that guarantee the existence of a positive periodic solution. Moreover, we provide sufficient conditions that assure the global attractivity of the positive periodic solution. Numerical results corroborate the theoretical results in the sense that the solutions are positive and the periodic solution is a global attractor. This type of models are important, since they take into account the variability of the weather and the impact on some epidemics such as the current COVID-19 pandemic.
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Affiliation(s)
- Abraham J. Arenas
- Departamento de Matemáticas y Estadística, Universidad de Córdoba, Montería, Colombia
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Periodically intermittent control for finite-time synchronization of delayed quaternion-valued neural networks. Neural Comput Appl 2020. [DOI: 10.1007/s00521-020-05417-1] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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10
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Xu X, Xu Q, Yang J, Xue H, Xu Y. Further research on exponential stability for quaternion-valued neural networks with mixed delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.03.004] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Li HL, Jiang H, Cao J. Global synchronization of fractional-order quaternion-valued neural networks with leakage and discrete delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2019.12.018] [Citation(s) in RCA: 37] [Impact Index Per Article: 9.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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12
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Almost periodic solutions of quaternion-valued neutral type high-order Hopfield neural networks with state-dependent delays and leakage delays. APPL INTELL 2020. [DOI: 10.1007/s10489-020-01634-2] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/05/2023]
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13
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Li Y, Wang C. Anti-periodic solutions for quaternion-valued inertial neural networks with time-varying delays1. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2019. [DOI: 10.3233/jifs-182731] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Yongkun Li
- Department of Mathematics, Yunnan University, Kunming, Yunnan, People’s Republic of China
| | - Chun Wang
- Department of Mathematics, Yunnan University, Kunming, Yunnan, People’s Republic of China
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Li Y, Meng X. Almost Automorphic Solutions in Distribution Sense of Quaternion-Valued Stochastic Recurrent Neural Networks with Mixed Time-Varying Delays. Neural Process Lett 2019. [DOI: 10.1007/s11063-019-10151-4] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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Huang C, Nie X, Zhao X, Song Q, Tu Z, Xiao M, Cao J. Novel bifurcation results for a delayed fractional-order quaternion-valued neural network. Neural Netw 2019; 117:67-93. [DOI: 10.1016/j.neunet.2019.05.002] [Citation(s) in RCA: 64] [Impact Index Per Article: 12.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/16/2019] [Revised: 05/04/2019] [Accepted: 05/06/2019] [Indexed: 11/16/2022]
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Li Y, Shen S. Almost automorphic solution of quaternion-valued BAM neural networks with time-varying delays on time scales1. JOURNAL OF INTELLIGENT & FUZZY SYSTEMS 2019. [DOI: 10.3233/jifs-181118] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
Affiliation(s)
- Yongkun Li
- Department of Mathematics, Yunnan University, Kunming, Yunnan, People’s Republic of China
| | - Shiping Shen
- Department of Mathematics, Yunnan University, Kunming, Yunnan, People’s Republic of China
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Ozcan N. Stability analysis of Cohen–Grossberg neural networks of neutral-type: Multiple delays case. Neural Netw 2019; 113:20-27. [DOI: 10.1016/j.neunet.2019.01.017] [Citation(s) in RCA: 34] [Impact Index Per Article: 6.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/19/2018] [Revised: 01/22/2019] [Accepted: 01/29/2019] [Indexed: 10/27/2022]
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Stability analysis of fractional Quaternion-Valued Leaky Integrator Echo State Neural Networks with multiple time-varying delays. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.11.021] [Citation(s) in RCA: 25] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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Meng X, Li Y. Pseudo almost periodic solutions for quaternion-valued cellular neural networks with discrete and distributed delays. JOURNAL OF INEQUALITIES AND APPLICATIONS 2018; 2018:245. [PMID: 30839647 PMCID: PMC6154083 DOI: 10.1186/s13660-018-1837-1] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 05/24/2018] [Accepted: 09/06/2018] [Indexed: 06/09/2023]
Abstract
This paper is concerned with a class of quaternion-valued cellular neural networks with discrete and distributed delays. By using the exponential dichotomy of linear systems and a fixed point theorem, sufficient conditions are derived for the existence and global exponential stability of pseudo almost periodic solutions of this class of neural networks. Finally, a numerical example is given to illustrate the feasibility of the obtained results.
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Affiliation(s)
- Xiaofang Meng
- Department of Mathematics, Yunnan University, Kunming, People’s Republic of China
| | - Yongkun Li
- Department of Mathematics, Yunnan University, Kunming, People’s Republic of China
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Li Y, Wang H. Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control. PLoS One 2018; 13:e0198297. [PMID: 29879145 PMCID: PMC5991723 DOI: 10.1371/journal.pone.0198297] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/19/2018] [Accepted: 05/16/2018] [Indexed: 11/20/2022] Open
Abstract
This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.
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Affiliation(s)
- Yongkun Li
- Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
- * E-mail:
| | - Huimei Wang
- Department of Mathematics, Yunnan University, Kunming, Yunnan 650091, China
- Department of Mathematics, Kunming University, Kunming, Yunnan 650214, China
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