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Liu P, Wang J, Zeng Z. An Overview of the Stability Analysis of Recurrent Neural Networks With Multiple Equilibria. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:1098-1111. [PMID: 34449396 DOI: 10.1109/tnnls.2021.3105519] [Citation(s) in RCA: 6] [Impact Index Per Article: 6.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
The stability analysis of recurrent neural networks (RNNs) with multiple equilibria has received extensive interest since it is a prerequisite for successful applications of RNNs. With the increasing theoretical results on this topic, it is desirable to review the results for a systematical understanding of the state of the art. This article provides an overview of the stability results of RNNs with multiple equilibria including complete stability and multistability. First, preliminaries on the complete stability and multistability analysis of RNNs are introduced. Second, the complete stability results of RNNs are summarized. Third, the multistability results of various RNNs are reviewed in detail. Finally, future directions in these interesting topics are suggested.
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2
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Yu T, Cao J, Rutkowski L, Luo YP. Finite-Time Synchronization of Complex-Valued Memristive-Based Neural Networks via Hybrid Control. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:3938-3947. [PMID: 33566775 DOI: 10.1109/tnnls.2021.3054967] [Citation(s) in RCA: 9] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
The finite-time synchronization problem is investigated for the master-slave complex-valued memristive neural networks in this article. A novel Lyapunov-function based finite-time stability criterion with impulsive effects is proposed and utilized to design the decentralized finite-time synchronization controller. Not only the settling time but also the attractive domain with respect to the impulsive gain and average impulsive interval, as well as initial values is derived according to the sufficient synchronization condition. Two examples are outlined to illustrate the validity of our hybrid control strategy.
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3
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Chen S, Song Q, Zhao Z, Liu Y, Alsaadi FE. Global asymptotic stability of fractional-order complex-valued neural networks with probabilistic time-varying delays. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.04.043] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/24/2022]
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4
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Zhang F, Huang T, Feng D, Zeng Z. Multistability and robustness of complex-valued neural networks with delays and input perturbation. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.03.045] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022]
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5
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Zhang J, Zhu S, Lu N, Wen S. Multistability of state-dependent switching neural networks with discontinuous nonmonotonic piecewise linear activation functions. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.01.046] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
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6
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Global exponential stability of delayed complex-valued neural networks with discontinuous activation functions. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.02.006] [Citation(s) in RCA: 12] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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7
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Liu P, Zheng WX, Zeng Z. On Complete Stability of Recurrent Neural Networks With Time-Varying Delays and General Piecewise Linear Activation Functions. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:2249-2263. [PMID: 30575557 DOI: 10.1109/tcyb.2018.2884836] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
This paper addresses the problem of complete stability of delayed recurrent neural networks with a general class of piecewise linear activation functions. By applying an appropriate partition of the state space and iterating the defined bounding functions, some sufficient conditions are obtained to ensure that an n -neuron neural network is completely stable with exactly ∏i=1n(2Ki-1) equilibrium points, among which ∏i=1nKi equilibrium points are locally exponentially stable and the others are unstable, where Ki (i=1,…,n) are non-negative integers which depend jointly on activation functions and parameters of neural networks. The results of this paper include the existing works on the stability analysis of recurrent neural networks with piecewise linear functions as special cases and hence can be considered as the improvement and extension of the existing stability results in the literature. A numerical example is provided to illustrate the derived theoretical results.
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8
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Finite time anti-synchronization of complex-valued neural networks with bounded asynchronous time-varying delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.01.035] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022]
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9
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Wang X, Wang Z, Song Q, Shen H, Huang X. A waiting-time-based event-triggered scheme for stabilization of complex-valued neural networks. Neural Netw 2020; 121:329-338. [DOI: 10.1016/j.neunet.2019.09.032] [Citation(s) in RCA: 34] [Impact Index Per Article: 8.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/25/2019] [Revised: 09/11/2019] [Accepted: 09/22/2019] [Indexed: 10/25/2022]
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10
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Liu D, Zhu S, Sun K. Global Anti-Synchronization of Complex-Valued Memristive Neural Networks With Time Delays. IEEE TRANSACTIONS ON CYBERNETICS 2019; 49:1735-1747. [PMID: 29993825 DOI: 10.1109/tcyb.2018.2812708] [Citation(s) in RCA: 23] [Impact Index Per Article: 4.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper formulates a class of complex-valued memristive neural networks as well as investigates the problem of anti-synchronization for complex-valued memristive neural networks. Under the concept of drive-response, several sufficient conditions for guaranteeing the anti-synchronization are given by employing suitable Lyapunov functional and some inequality techniques. The proposed results of this paper are less conservative than existing literatures due to the characteristics of memristive complex-valued neural networks. Moreover, the proposed results are easy to be validated with the parameters of system itself. Finally, two examples with numerical simulations are showed to demonstrate the efficiency of our theoretical results.
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Liu P, Nie X, Liang J, Cao J. Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions. Neural Netw 2018; 108:452-465. [DOI: 10.1016/j.neunet.2018.09.005] [Citation(s) in RCA: 43] [Impact Index Per Article: 7.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/08/2018] [Revised: 09/05/2018] [Accepted: 09/07/2018] [Indexed: 11/28/2022]
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12
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Wang Z, Cao J, Guo Z, Huang L. Generalized stability for discontinuous complex-valued Hopfield neural networks via differential inclusions. Proc Math Phys Eng Sci 2018; 474:20180507. [PMID: 30602935 PMCID: PMC6304032 DOI: 10.1098/rspa.2018.0507] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/31/2018] [Accepted: 10/24/2018] [Indexed: 11/12/2022] Open
Abstract
Some dynamical behaviours of discontinuous complex-valued Hopfield neural networks are discussed in this paper. First, we introduce a method to construct the complex-valued set-valued mapping and define some basic definitions for discontinuous complex-valued differential equations. In addition, Leray-Schauder alternative theorem is used to analyse the equilibrium existence of the networks. Lastly, we present the dynamical behaviours, including global stability and convergence in measure for discontinuous complex-valued neural networks (CVNNs) via differential inclusions. The main contribution of this paper is that we extend previous studies on continuous CVNNs to discontinuous ones. Several simulations are given to substantiate the correction of the proposed results.
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Affiliation(s)
- Zengyun Wang
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, Southeast University, Nanjing 210096, People's Republic of China
- School of Mathematics, Southeast University, Nanjing 210096, People's Republic of China
- Department of Mathematics, Hunan First Normal University, Changsha 410205, People's Republic of China
| | - Jinde Cao
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, Southeast University, Nanjing 210096, People's Republic of China
- School of Mathematics, Southeast University, Nanjing 210096, People's Republic of China
| | - Zhenyuan Guo
- College of Mathematics and Economtrics, Hunan University, Changsha 410082, People's Republic of China
| | - Lihong Huang
- Changsha University of Science and Technology, Changsha 410114, People's Republic of China
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13
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Song Q, Chen X. Multistability Analysis of Quaternion-Valued Neural Networks With Time Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2018; 29:5430-5440. [PMID: 29994739 DOI: 10.1109/tnnls.2018.2801297] [Citation(s) in RCA: 69] [Impact Index Per Article: 11.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
This paper addresses the multistability issue for quaternion-valued neural networks (QVNNs) with time delays. By using the inequality technique, sufficient conditions are proposed for the boundedness and the global attractivity of delayed QVNNs. Based on the geometrical properties of the activation functions, several criteria are obtained to ensure the existence of equilibrium points, of which are locally stable. Two numerical examples are provided to illustrate the effectiveness of the obtained results.
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14
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Liu P, Zeng Z, Wang J. Multistability of Recurrent Neural Networks With Nonmonotonic Activation Functions and Unbounded Time-Varying Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2018; 29:3000-3010. [PMID: 28678718 DOI: 10.1109/tnnls.2017.2710299] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
This paper is concerned with the coexistence of multiple equilibrium points and dynamical behaviors of recurrent neural networks with nonmonotonic activation functions and unbounded time-varying delays. Based on a state space partition by using the geometrical properties of the activation functions, it is revealed that an -neuron neural network can exhibit equilibrium points with . In particular, several sufficient criteria are proposed to ascertain the asymptotical stability of equilibrium points for recurrent neural networks. These theoretical results cover both monostability and multistability. Furthermore, the attraction basins of asymptotically stable equilibrium points are estimated. It is shown that the attraction basins of the stable equilibrium points can be larger than their originally partitioned subsets. Finally, the results are illustrated by using the simulation results of four examples.
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15
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Marco MD, Forti M, Grazzini M, Pancioni L. Multistability of delayed neural networks with hard-limiter saturation nonlinearities. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2018.03.006] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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16
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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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17
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Li L, Wang Z, Lu J, Li Y. Adaptive Synchronization of Fractional-Order Complex-Valued Neural Networks with Discrete and Distributed Delays. ENTROPY (BASEL, SWITZERLAND) 2018; 20:E124. [PMID: 33265215 PMCID: PMC7512617 DOI: 10.3390/e20020124] [Citation(s) in RCA: 18] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 01/25/2018] [Revised: 02/10/2018] [Accepted: 02/11/2018] [Indexed: 11/22/2022]
Abstract
In this paper, the synchronization problem of fractional-order complex-valued neural networks with discrete and distributed delays is investigated. Based on the adaptive control and Lyapunov function theory, some sufficient conditions are derived to ensure the states of two fractional-order complex-valued neural networks with discrete and distributed delays achieve complete synchronization rapidly. Finally, numerical simulations are given to illustrate the effectiveness and feasibility of the theoretical results.
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Affiliation(s)
- Li Li
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
| | - Zhen Wang
- College of Mathematics and Systems Science, Shandong University of Science and Technology, Qingdao 266590, China
- College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
| | - Junwei Lu
- School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 210023, China
| | - Yuxia Li
- College of Electrical Engineering and Automation, Shandong University of Science and Technology, Qingdao 266590, China
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18
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Tan M, Xu D. Multiple μ-stability analysis for memristor-based complex-valued neural networks with nonmonotonic piecewise nonlinear activation functions and unbounded time-varying delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.11.047] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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19
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New results for exponential stability of complex-valued memristive neural networks with variable delays. Neurocomputing 2018. [DOI: 10.1016/j.neucom.2017.08.066] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/20/2022]
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20
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Liu D, Zhu S, Ye E. Synchronization stability of memristor-based complex-valued neural networks with time delays. Neural Netw 2017; 96:115-127. [PMID: 28987975 DOI: 10.1016/j.neunet.2017.09.008] [Citation(s) in RCA: 40] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/16/2017] [Revised: 08/30/2017] [Accepted: 09/01/2017] [Indexed: 11/16/2022]
Abstract
This paper focuses on the dynamical property of a class of memristor-based complex-valued neural networks (MCVNNs) with time delays. By constructing the appropriate Lyapunov functional and utilizing the inequality technique, sufficient conditions are proposed to guarantee exponential synchronization of the coupled systems based on drive-response concept. The proposed results are very easy to verify, and they also extend some previous related works on memristor-based real-valued neural networks. Meanwhile, the obtained sufficient conditions of this paper may be conducive to qualitative analysis of some complex-valued nonlinear delayed systems. A numerical example is given to demonstrate the effectiveness of our theoretical results.
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Affiliation(s)
- Dan Liu
- School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China
| | - Song Zhu
- School of Mathematics, China University of Mining and Technology, Xuzhou, 221116, China.
| | - Er Ye
- School of Mathematics, Southeast University, Nanjing 210096, China
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21
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Hu J, Zeng C. Adaptive exponential synchronization of complex-valued Cohen–Grossberg neural networks with known and unknown parameters. Neural Netw 2017; 86:90-101. [DOI: 10.1016/j.neunet.2016.11.001] [Citation(s) in RCA: 54] [Impact Index Per Article: 7.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2016] [Revised: 09/24/2016] [Accepted: 11/04/2016] [Indexed: 11/25/2022]
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22
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Dynamical Behavior of Complex-Valued Hopfield Neural Networks with Discontinuous Activation Functions. Neural Process Lett 2016. [DOI: 10.1007/s11063-016-9563-5] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/20/2022]
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23
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Liang J, Gong W, Huang T. Multistability of complex-valued neural networks with discontinuous activation functions. Neural Netw 2016; 84:125-142. [PMID: 27718391 DOI: 10.1016/j.neunet.2016.08.008] [Citation(s) in RCA: 61] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/30/2016] [Revised: 06/21/2016] [Accepted: 08/23/2016] [Indexed: 11/29/2022]
Abstract
In this paper, based on the geometrical properties of the discontinuous activation functions and the Brouwer's fixed point theory, the multistability issue is tackled for the complex-valued neural networks with discontinuous activation functions and time-varying delays. To address the network with discontinuous functions, Filippov solution of the system is defined. Through rigorous analysis, several sufficient criteria are obtained to assure the existence of 25n equilibrium points. Among them, 9n points are locally stable and 16n-9n equilibrium points are unstable. Furthermore, to enlarge the attraction basins of the 9n equilibrium points, some mild conditions are imposed. Finally, one numerical example is provided to illustrate the effectiveness of the obtained results.
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Affiliation(s)
- Jinling Liang
- Department of Mathematics, Southeast University, Nanjing 210096, China.
| | - Weiqiang Gong
- Department of Mathematics, Southeast University, Nanjing 210096, China.
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24
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Xie D, Jiang Y. Global exponential stability of periodic solution for delayed complex-valued neural networks with impulses. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2016.04.054] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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25
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Synchronization of fractional-order complex-valued neural networks with time delay. Neural Netw 2016; 81:16-28. [DOI: 10.1016/j.neunet.2016.05.003] [Citation(s) in RCA: 191] [Impact Index Per Article: 23.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/25/2015] [Revised: 03/20/2016] [Accepted: 05/09/2016] [Indexed: 11/23/2022]
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26
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Global exponential stability of impulsive complex-valued neural networks with both asynchronous time-varying and continuously distributed delays. Neural Netw 2016; 81:1-10. [DOI: 10.1016/j.neunet.2016.04.012] [Citation(s) in RCA: 140] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/03/2016] [Revised: 04/25/2016] [Accepted: 04/29/2016] [Indexed: 11/22/2022]
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27
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Xu D, Dong J, Zhang H. Deterministic Convergence of Wirtinger-Gradient Methods for Complex-Valued Neural Networks. Neural Process Lett 2016. [DOI: 10.1007/s11063-016-9535-9] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/03/2023]
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28
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Global exponential stability of complex-valued neural networks with both time-varying delays and impulsive effects. Neural Netw 2016; 79:108-16. [DOI: 10.1016/j.neunet.2016.03.007] [Citation(s) in RCA: 164] [Impact Index Per Article: 20.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/15/2015] [Revised: 01/30/2016] [Accepted: 03/17/2016] [Indexed: 11/19/2022]
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29
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Liu P, Zeng Z, Wang J. Multistability analysis of a general class of recurrent neural networks with non-monotonic activation functions and time-varying delays. Neural Netw 2016; 79:117-27. [DOI: 10.1016/j.neunet.2016.03.010] [Citation(s) in RCA: 46] [Impact Index Per Article: 5.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/01/2015] [Revised: 03/21/2016] [Accepted: 03/29/2016] [Indexed: 11/29/2022]
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30
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Gong W, Liang J, Zhang C. Multistability of complex-valued neural networks with distributed delays. Neural Comput Appl 2016. [DOI: 10.1007/s00521-016-2305-9] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/21/2022]
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31
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Chen X, Song Q, Zhao Z, Liu Y. Global μ -stability analysis of discrete-time complex-valued neural networks with leakage delay and mixed delays. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2015.10.120] [Citation(s) in RCA: 19] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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32
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Li Y, Liao X, Li H. Global attracting sets of non-autonomous and complex-valued neural networks with time-varying delays. Neurocomputing 2016. [DOI: 10.1016/j.neucom.2015.08.056] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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33
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Gong W, Liang J, Cao J. Global μ-stability of complex-valued delayed neural networks with leakage delay. Neurocomputing 2015. [DOI: 10.1016/j.neucom.2015.06.006] [Citation(s) in RCA: 31] [Impact Index Per Article: 3.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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34
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Global exponential periodicity and stability of discrete-time complex-valued recurrent neural networks with time-delays. Neural Netw 2015; 66:119-30. [DOI: 10.1016/j.neunet.2015.03.001] [Citation(s) in RCA: 64] [Impact Index Per Article: 7.1] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/20/2014] [Revised: 02/17/2015] [Accepted: 03/03/2015] [Indexed: 11/22/2022]
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