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Cao J, Udhayakumar K, Rakkiyappan R, Li X, Lu J. A Comprehensive Review of Continuous-/Discontinuous-Time Fractional-Order Multidimensional Neural Networks. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:5476-5496. [PMID: 34962883 DOI: 10.1109/tnnls.2021.3129829] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/14/2023]
Abstract
The dynamical study of continuous-/discontinuous-time fractional-order neural networks (FONNs) has been thoroughly explored, and several publications have been made available. This study is designed to give an exhaustive review of the dynamical studies of multidimensional FONNs in continuous/discontinuous time, including Hopfield NNs (HNNs), Cohen-Grossberg NNs, and bidirectional associative memory NNs, and similar models are considered in real ( [Formula: see text]), complex ( [Formula: see text]), quaternion ( [Formula: see text]), and octonion ( [Formula: see text]) fields. Since, in practice, delays are unavoidable, theoretical findings from multidimensional FONNs with various types of delays are thoroughly evaluated. Some required and adequate stability and synchronization requirements are also mentioned for fractional-order NNs without delays.
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Li B, Cheng X. Synchronization analysis of coupled fractional-order neural networks with time-varying delays. MATHEMATICAL BIOSCIENCES AND ENGINEERING : MBE 2023; 20:14846-14865. [PMID: 37679162 DOI: 10.3934/mbe.2023665] [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, the complete synchronization and Mittag-Leffler synchronization problems of a kind of coupled fractional-order neural networks with time-varying delays are introduced and studied. First, the sufficient conditions for a controlled system to reach complete synchronization are established by using the Kronecker product technique and Lyapunov direct method under pinning control. Here the pinning controller only needs to control part of the nodes, which can save more resources. To make the system achieve complete synchronization, only the error system is stable. Next, a new adaptive feedback controller is designed, which combines the Razumikhin-type method and Mittag-Leffler stability theory to make the controlled system realize Mittag-Leffler synchronization. The controller has time delays, and the calculation can be simplified by constructing an appropriate auxiliary function. Finally, two numerical examples are given. The simulation process shows that the conditions of the main theorems are not difficult to obtain, and the simulation results confirm the feasibility of the theorems.
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Affiliation(s)
- Biwen Li
- School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
| | - Xuan Cheng
- School of Mathematics and Statistics, Hubei Normal University, Huangshi 435002, China
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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.
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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.
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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.
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Bipartite leader-following synchronization of delayed incommensurate fractional-order memristor-based neural networks under signed digraph via adaptive strategy. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2022.06.043] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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Chen J, Chen B, Zeng Z, Jiang P. Event-Based Synchronization for Multiple Neural Networks With Time Delay and Switching Disconnected Topology. IEEE TRANSACTIONS ON CYBERNETICS 2021; 51:5993-6003. [PMID: 31976921 DOI: 10.1109/tcyb.2019.2960762] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
This article discusses the synchronization problem for a class of multiple delayed neural networks (MDNNs) with a directed switching topology by using an event-triggering strategy. First, a new differential inequality with delay is shown, which is a generalization of Halanay-type inequalities. Then, the sufficient conditions of event-based synchronization (quasisynchronization) for MDNN with sequentially connected topology are obtained by using this inequality and the iterative method. Meantime, we prove that Zeno behavior can be avoided under the designed event-triggering rules. As an extension, MDNN with jointly connected topology is also discussed. Finally, a numerical example is listed to illustrate the results in theory analysis.
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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]
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Zheng B, Hu C, Yu J, Jiang H. Synchronization analysis for delayed spatio-temporal neural networks with fractional-order. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.01.128] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Zhang F, Huang T, Wu Q, Zeng Z. Multistability of delayed fractional-order competitive neural networks. Neural Netw 2021; 140:325-335. [PMID: 33895556 DOI: 10.1016/j.neunet.2021.03.036] [Citation(s) in RCA: 11] [Impact Index Per Article: 3.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/04/2020] [Revised: 02/27/2021] [Accepted: 03/24/2021] [Indexed: 10/21/2022]
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.
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Affiliation(s)
- Fanghai Zhang
- School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China; Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China.
| | - Tingwen Huang
- Science Program, Texas A&M University at Qatar, Doha, Qatar.
| | - Qiujie Wu
- School of Internet, Anhui University, Hefei, China.
| | - Zhigang Zeng
- School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, China; Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan, China.
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Jia J, Zeng Z. LMI-based criterion for global Mittag-Leffler lag quasi-synchronization of fractional-order memristor-based neural networks via linear feedback pinning control. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.05.074] [Citation(s) in RCA: 8] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/24/2022]
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Finite-Time Mittag–Leffler Synchronization of Neutral-Type Fractional-Order Neural Networks with Leakage Delay and Time-Varying Delays. MATHEMATICS 2020. [DOI: 10.3390/math8071146] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This paper studies fractional-order neural networks with neutral-type delay, leakage delay, and time-varying delays. A sufficient condition which ensures the finite-time synchronization of these networks based on a state feedback control scheme is deduced using the generalized Gronwall–Bellman inequality. Then, a different state feedback control scheme is employed to realize the finite-time Mittag–Leffler synchronization of these networks by using the fractional-order extension of the Lyapunov direct method for Mittag–Leffler stability. Two numerical examples illustrate the feasibility and the effectiveness of the deduced sufficient criteria.
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Chen J, Chen B, Zeng Z, Jiang P. Event-Triggered Synchronization Strategy for Multiple Neural Networks With Time Delay. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:3271-3280. [PMID: 31034433 DOI: 10.1109/tcyb.2019.2911029] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
This paper deals with global exponential synchronization of multiple neural networks (NNs) with time delay via a very broad class of event-triggered coupling, in which coupling matrix can be non-Laplacian. Some simple and convenient sufficient conditions are derived to guarantee global exponential synchronization of the coupling NNs under an event-triggered strategy. In particular, the effect of the common subsystem can be positive or negative on the synchronization scheme. Three examples are presented to test the results in theory analysis.
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Lv Y, Hu C, Yu J, Jiang H, Huang T. Edge-Based Fractional-Order Adaptive Strategies for Synchronization of Fractional-Order Coupled Networks With Reaction-Diffusion Terms. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:1582-1594. [PMID: 30507521 DOI: 10.1109/tcyb.2018.2879935] [Citation(s) in RCA: 15] [Impact Index Per Article: 3.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
In this paper, spatial diffusions are introduced to fractional-order coupled networks and the problem of synchronization is investigated for fractional-order coupled neural networks with reaction-diffusion terms. First, a new fractional-order inequality is established based on the Caputo partial fractional derivative. To realize asymptotical synchronization, two types of adaptive coupling weights are considered, namely: 1) coupling weights only related to time and 2) coupling weights dependent on both time and space. For each type of coupling weights, based on local information of the node's dynamics, an edge-based fractional-order adaptive law and an edge-based fractional-order pinning adaptive scheme are proposed. Furthermore, some new analytical tools, including the method of contradiction, L'Hopital rule, and Barbalat lemma are developed to establish adaptive synchronization criteria of the addressed networks. Finally, an example with numerical simulations is provided to illustrate the validity and effectiveness of the theoretical results.
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Li Y, Luo B, Liu D, Yang Z, Zhu Y. Adaptive synchronization of memristor-based neural networks with discontinuous activations. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2019.11.018] [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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Chen J, Chen B, Zeng Z, Jiang P. Effects of Subsystem and Coupling on Synchronization of Multiple Neural Networks With Delays via Impulsive Coupling. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:3748-3758. [PMID: 30892235 DOI: 10.1109/tnnls.2019.2898919] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
This paper from new perspectives discusses the global synchronization of multiple recurrent neural networks (MNNs) with time delays via impulsive coupling. A new concept (coupling strength) is introduced, it is a variable parameter and plays a key role on synchronization. The selection of coupling strength can bring more convenience to the design of the impulsive coupling controller. Four results are presented for the synchronization of MNNs with time delays by using impulsive coupling with the coupling gain and variable topology, where two results are dependent on topology and other two results are independent on topological connectivity. In our results, the effects of each NN, coupling topology, and coupling strength can be positive or negative role on synchronization. In addition, three examples are presented to test our results in the theory analysis.
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Chen L, Huang T, Tenreiro Machado J, Lopes AM, Chai Y, Wu R. Delay-dependent criterion for asymptotic stability of a class of fractional-order memristive neural networks with time-varying delays. Neural Netw 2019; 118:289-299. [DOI: 10.1016/j.neunet.2019.07.006] [Citation(s) in RCA: 46] [Impact Index Per Article: 9.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/07/2019] [Revised: 06/04/2019] [Accepted: 07/07/2019] [Indexed: 11/16/2022]
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