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Xu C, Lin J, Zhao Y, Cui Q, Ou W, Pang Y, Liu Z, Liao M, Li P. New results on bifurcation for fractional-order octonion-valued neural networks involving delays. NETWORK (BRISTOL, ENGLAND) 2024:1-53. [PMID: 38578214 DOI: 10.1080/0954898x.2024.2332662] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/17/2023] [Accepted: 03/09/2024] [Indexed: 04/06/2024]
Abstract
This work chiefly explores fractional-order octonion-valued neural networks involving delays. We decompose the considered fractional-order delayed octonion-valued neural networks into equivalent real-valued systems via Cayley-Dickson construction. By virtue of Lipschitz condition, we prove that the solution of the considered fractional-order delayed octonion-valued neural networks exists and is unique. By constructing a fairish function, we confirm that the solution of the involved fractional-order delayed octonion-valued neural networks is bounded. Applying the stability theory and basic bifurcation knowledge of fractional order differential equations, we set up a sufficient condition remaining the stability behaviour and the appearance of Hopf bifurcation for the addressed fractional-order delayed octonion-valued neural networks. To illustrate the justifiability of the derived theoretical results clearly, we give the related simulation results to support these facts. Simultaneously, the bifurcation plots are also displayed. The established theoretical results in this work have important guiding significance in devising and improving neural networks.
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Affiliation(s)
- Changjin Xu
- Guizhou Key Laboratory of Economics System Simulation, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Jinting Lin
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Yingyan Zhao
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Qingyi Cui
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Wei Ou
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Yicheng Pang
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Zixin Liu
- School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang, P.R. China
| | - Maoxin Liao
- School of Mathematics and Physics, University of South China, Hengyang, P.R. China
| | - Peiluan Li
- School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, P.R. China
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Xia Z, Liu Y, Kou KI, Wang J. Clifford-Valued Distributed Optimization Based on Recurrent Neural Networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:7248-7259. [PMID: 35030085 DOI: 10.1109/tnnls.2021.3139865] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
In this paper, we address the Clifford-valued distributed optimization subject to linear equality and inequality constraints. The objective function of the optimization problems is composed of the sum of convex functions defined in the Clifford domain. Based on the generalized Clifford gradient, a system of multiple Clifford-valued recurrent neural networks (RNNs) is proposed for solving the distributed optimization problems. Each Clifford-valued RNN minimizes a local objective function individually, with local interactions with others. The convergence of the neural system is rigorously proved based on the Lyapunov theory. Two illustrative examples are delineated to demonstrate the viability of the results in this article.
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Anti-periodic solutions on Clifford-valued high-order Hopfield neural networks with multi-proportional delays. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2021.11.001] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/15/2022]
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4
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Existence and Global Attractivity of Pseudo Almost Periodic Solutions for Clifford-Valued Fuzzy Neural Networks with Proportional Delays. MATHEMATICS 2021. [DOI: 10.3390/math9243306] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
In this paper, Clifford-valued fuzzy neural networks with proportional delays, whose leakage term coefficients are also Clifford numbers, are considered. Based on the Banach fixed point theorem and differential inequality technique, we use a direct method to obtain the existence, uniqueness, and global attractivity of pseudo almost periodic solutions for the considered networks. Finally, we provide a numerical example to illustrate the feasibility of our results. Our results are new.
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Synchronization in Fixed Time for Reaction–Diffusion Quaternion-Valued NNs with Nonlinear Interconnected Protocol and Its Application. Neural Process Lett 2021. [DOI: 10.1007/s11063-021-10579-7] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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Synchronization in Finite-Time Analysis of Clifford-Valued Neural Networks with Finite-Time Distributed Delays. MATHEMATICS 2021. [DOI: 10.3390/math9111163] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
Abstract
In this paper, we explore the finite-time synchronization of Clifford-valued neural networks with finite-time distributed delays. To address the problem associated with non-commutativity pertaining to the multiplication of Clifford numbers, the original n-dimensional Clifford-valued drive and response systems are firstly decomposed into the corresponding 2m-dimensional real-valued counterparts. On the basis of a new Lyapunov–Krasovskii functional, suitable controller and new computational techniques, finite-time synchronization criteria are formulated for the corresponding real-valued drive and response systems. The feasibility of the main results is verified by a numerical example.
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Li Y, Huo N, Li B. On μ-Pseudo Almost Periodic Solutions for Clifford-Valued Neutral Type Neural Networks With Delays in the Leakage Term. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2021; 32:1365-1374. [PMID: 32305945 DOI: 10.1109/tnnls.2020.2984655] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
We propose a class of Clifford-valued neutral type neural networks with delays in the leakage term. Using a direct method, that is, without decomposing the Clifford-valued system under consideration into a real-valued system, we obtain sufficient conditions for the existence and global exponential stability of μ -pseudo almost periodic solutions of the Clifford-valued neural network under consideration. Finally, we give a numerical example to show the feasibility of our results.
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Aouiti C, Dridi F, Hui Q, Moulay E. $$(\mu ,\nu )-$$Pseudo Almost Automorphic Solutions of Neutral Type Clifford-Valued High-Order Hopfield Neural Networks with D Operator. Neural Process Lett 2021. [DOI: 10.1007/s11063-020-10421-6] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Li Y, Shen S. Almost automorphic solutions for Clifford-valued neutral-type fuzzy cellular neural networks with leakage delays on time scales. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.07.035] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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Li Y, Huo N. (μ,ν)-pseudo almost periodic solutions of Clifford-valued high-order HNNs with multiple discrete delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.07.069] [Citation(s) in RCA: 9] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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Popa CA. Dissipativity of impulsive matrix-valued neural networks with leakage delay and mixed delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.03.042] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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$$S^{p}$$-Almost Periodic Solutions of Clifford-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays. Neural Process Lett 2020. [DOI: 10.1007/s11063-019-10176-9] [Citation(s) in RCA: 24] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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Li Y, Xiang J. Existence and global exponential stability of anti-periodic solution for Clifford-valued inertial Cohen–Grossberg neural networks with delays. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.064] [Citation(s) in RCA: 37] [Impact Index Per Article: 7.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/27/2022]
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Popa CA. Global exponential stability of neutral-type octonion-valued neural networks with time-varying delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.05.004] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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Global exponential stability of octonion-valued neural networks with leakage delay and mixed delays. Neural Netw 2018; 105:277-293. [DOI: 10.1016/j.neunet.2018.05.006] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2017] [Revised: 03/12/2018] [Accepted: 05/04/2018] [Indexed: 11/19/2022]
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Chen X, Song Q, Li Z, Zhao Z, Liu Y. Stability Analysis of Continuous-Time and Discrete-Time Quaternion-Valued Neural Networks With Linear Threshold Neurons. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2018; 29:2769-2781. [PMID: 28600263 DOI: 10.1109/tnnls.2017.2704286] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This paper addresses the problem of stability for continuous-time and discrete-time quaternion-valued neural networks (QVNNs) with linear threshold neurons. Applying the semidiscretization technique to the continuous-time QVNNs, the discrete-time analogs are obtained, which preserve the dynamical characteristics of their continuous-time counterparts. Via the plural decomposition method of quaternion, homeomorphic mapping theorem, as well as Lyapunov theorem, some sufficient conditions on the existence, uniqueness, and global asymptotical stability of the equilibrium point are derived for the continuous-time QVNNs and their discrete-time analogs, respectively. Furthermore, a uniform sufficient condition on the existence, uniqueness, and global asymptotical stability of the equilibrium point is obtained for both continuous-time QVNNs and their discrete-time version. Finally, two numerical examples are provided to substantiate the effectiveness of the proposed results.
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Invariant and Attracting Sets of Complex-Valued Neural Networks with Both Time-Varying and Infinite Distributed Delays. Neural Process Lett 2018. [DOI: 10.1007/s11063-018-9848-y] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/26/2022]
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18
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Popa CA, Kaslik E. Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays. Neural Netw 2018; 99:1-18. [DOI: 10.1016/j.neunet.2017.12.006] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/10/2017] [Revised: 10/25/2017] [Accepted: 12/12/2017] [Indexed: 10/18/2022]
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Chen X, Li Z, Song Q, Hu J, Tan Y. Robust stability analysis of quaternion-valued neural networks with time delays and parameter uncertainties. Neural Netw 2017; 91:55-65. [DOI: 10.1016/j.neunet.2017.04.006] [Citation(s) in RCA: 111] [Impact Index Per Article: 15.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/01/2016] [Revised: 02/17/2017] [Accepted: 04/14/2017] [Indexed: 11/30/2022]
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Shu H, Song Q, Liu Y, Zhao Z, Alsaadi FE. Globalμ−stability of quaternion-valued neural networks with non-differentiable time-varying delays. Neurocomputing 2017. [DOI: 10.1016/j.neucom.2017.03.052] [Citation(s) in RCA: 69] [Impact Index Per Article: 9.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
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