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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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Guo Z, Ou S, Wang J. Multistability of Switched Neural Networks With Gaussian Activation Functions Under State-Dependent Switching. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2022; 33:6569-6583. [PMID: 34077372 DOI: 10.1109/tnnls.2021.3082560] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
This article presents theoretical results on the multistability of switched neural networks with Gaussian activation functions under state-dependent switching. It is shown herein that the number and location of the equilibrium points of the switched neural networks can be characterized by making use of the geometrical properties of Gaussian functions and local linearization based on the Brouwer fixed-point theorem. Four sets of sufficient conditions are derived to ascertain the existence of 7p15p23p3 equilibrium points, and 4p13p22p3 of them are locally stable, wherein p1 , p2 , and p3 are nonnegative integers satisfying 0 ≤ p1+p2+p3 ≤ n and n is the number of neurons. It implies that there exist up to 7n equilibria, and up to 4n of them are locally stable when p1=n . It also implies that properly selecting p1 , p2 , and p3 can engender a desirable number of stable equilibria. Two numerical examples are elaborated to substantiate the theoretical results.
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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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Zhang F, Zeng Z. Multiple Mittag-Leffler Stability of Delayed Fractional-Order Cohen-Grossberg Neural Networks via Mixed Monotone Operator Pair. IEEE TRANSACTIONS ON CYBERNETICS 2021; 51:6333-6344. [PMID: 31995512 DOI: 10.1109/tcyb.2019.2963034] [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 mainly investigates the multiple Mittag-Leffler stability of delayed fractional-order Cohen-Grossberg neural networks with time-varying delays. By using mixed monotone operator pair, the conditions of the coexistence of multiple equilibrium points are obtained for fractional-order Cohen-Grossberg neural networks, and these conditions are eventually transformed into algebraic inequalities based on the vertex of the divided region. In particular, when the symbols of these inequalities are determined by the dominant term, several verifiable corollaries are given. And then, the sufficient conditions of the Mittag-Leffler stability are derived for fractional-order Cohen-Grossberg neural networks with time-varying delays. In addition, two numerical examples are provided to illustrate the effectiveness of the theoretical results.
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Multiple asymptotic stability of fractional-order quaternion-valued neural networks with time-varying delays. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.03.079] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/17/2022]
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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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Zhang F, Zeng Z. Robust Stability of Recurrent Neural Networks With Time-Varying Delays and Input Perturbation. IEEE TRANSACTIONS ON CYBERNETICS 2021; 51:3027-3038. [PMID: 31329152 DOI: 10.1109/tcyb.2019.2926537] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
This paper addresses the robust stability of recurrent neural networks (RNNs) with time-varying delays and input perturbation, where the time-varying delays include discrete and distributed delays. By employing the new ψ -type integral inequality, several sufficient conditions are derived for the robust stability of RNNs with discrete and distributed delays. Meanwhile, the robust boundedness of neural networks is explored by the bounded input perturbation and L1 -norm constraint. Moreover, RNNs have a strong anti-jamming ability to input perturbation, and the robustness of RNNs is suitable for associative memory. Specifically, when input perturbation belongs to the specified and well-characterized space, the results cover both monostability and multistability as special cases. It is revealed that there is a relationship between the stability of neural networks and input perturbation. Compared with the existing results, these conditions proposed in this paper improve and extend the existing stability in some literature. Finally, the numerical examples are given to substantiate the effectiveness of the theoretical results.
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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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Shen Y, Zhu S. Multistability of delayed neural networks with monotonically nondecreasing linear activation function. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2020.10.011] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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Zhang F, Zeng Z. Multistability of Fractional-Order Neural Networks With Unbounded Time-Varying Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2021; 32:177-187. [PMID: 32203030 DOI: 10.1109/tnnls.2020.2977994] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
This article addresses the multistability and attraction of fractional-order neural networks (FONNs) with unbounded time-varying delays. Several sufficient conditions are given to ensure the coexistence of equilibrium points (EPs) of FONNs with concave-convex activation functions. Moreover, by exploiting the analytical method and the property of the Mittag-Leffler function, it is shown that the multiple Mittag-Leffler stability of delayed FONNs is derived and the obtained criteria do not depend on differentiable time-varying delays. In particular, the criterion of the Mittag-Leffler stability can be simplified to M-matrix. In addition, the estimation of attraction basin of delayed FONNs is studied, which implies that the extension of attraction basin is independent of the magnitude of delays. Finally, three numerical examples are given to show the validity of the theoretical results.
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Guo Z, Ou S, Wang J. Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching. Neural Netw 2020; 122:239-252. [DOI: 10.1016/j.neunet.2019.10.012] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/28/2019] [Revised: 09/04/2019] [Accepted: 10/17/2019] [Indexed: 11/12/2022]
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Wan P, Sun D, Zhao M, Wan L, Jin S. Multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions. Neural Netw 2019; 122:231-238. [PMID: 31698201 DOI: 10.1016/j.neunet.2019.10.005] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/05/2019] [Revised: 08/20/2019] [Accepted: 10/08/2019] [Indexed: 11/26/2022]
Abstract
This paper is concerned with multistability and attraction basins of discrete-time neural networks with nonmonotonic piecewise linear activation functions. Under some reasonable conditions, the addressed networks have (2m+1)n equilibrium points. (m+1)n of which are locally asymptotically stable, and the others are unstable. The attraction basins of the locally asymptotically stable equilibrium points are given in the form of hyperspherical regions. These results here, which include existence, uniqueness, locally asymptotical stability, instability and attraction basins of the multiple equilibrium points, generalize and improve the earlier publications. Finally, an illustrative example with numerical simulation is given to show the feasibility and the effectiveness of the theoretical results. The theoretical results and illustrative example indicate that the activation functions improve the storage capacity of neural networks significantly.
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Affiliation(s)
- Peng Wan
- Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, Chongqing University, Chongqing 400044, China; School of Automation, Chongqing University, Chongqing 400044, China
| | - Dihua Sun
- Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, Chongqing University, Chongqing 400044, China; School of Automation, Chongqing University, Chongqing 400044, China.
| | - Min Zhao
- Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, Chongqing University, Chongqing 400044, China; School of Automation, Chongqing University, Chongqing 400044, China
| | - Li Wan
- School of Software Engineering, Southeast University, Nanjing, 211189, China
| | - Shuang Jin
- Key Laboratory of Dependable Service Computing in Cyber Physical Society of Ministry of Education, Chongqing University, Chongqing 400044, China; School of Automation, Chongqing University, Chongqing 400044, China
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Zhang F, Zeng Z. Multiple ψ -Type Stability and Its Robustness for Recurrent Neural Networks With Time-Varying Delays. IEEE TRANSACTIONS ON CYBERNETICS 2019; 49:1803-1815. [PMID: 29993797 DOI: 10.1109/tcyb.2018.2813979] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
In this paper, the ψ -type stability and robustness of recurrent neural networks are investigated by using the differential inequality. By utilizing ψ -type functions combined with the inequality techniques, some sufficient conditions ensuring ψ -type stability and robustness are derived for linear neural networks with time-varying delays. Then, by choosing appropriate Lipschitz coefficient in subregion, some algebraic criteria of the multiple ψ -type stability and robust boundedness are established for the delayed neural networks with time-varying delays. For special cases, several criteria are also presented by selecting parameters with easy implementation. The derived results cover both ψ -type mono-stability and multiple ψ -type stability. In addition, these theoretical results contain exponential stability, polynomial stability, and μ -stability, and they also complement and extend some previous results. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed criteria.
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Guo T, Wang L, Zhou M, Duan S. A multi-layer memristive recurrent neural network for solving static and dynamic image associative memory. Neurocomputing 2019. [DOI: 10.1016/j.neucom.2018.12.056] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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16
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Zhang F, Zeng Z. Multiple ψ -Type Stability of Cohen-Grossberg Neural Networks With Both Time-Varying Discrete Delays and Distributed Delays. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2019; 30:566-579. [PMID: 29994620 DOI: 10.1109/tnnls.2018.2846249] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
In this paper, multiple ψ -type stability of Cohen-Grossberg neural networks (CGNNs) with both time-varying discrete delays and distributed delays is investigated. By utilizing ψ -type functions combined with a new ψ -type integral inequality for treating distributed delay terms, some sufficient conditions are obtained to ensure that multiple equilibrium points are ψ -type stable for CGNNs with discrete and distributed delays, where the distributed delays include bounded and unbounded delays. These conditions of CGNNs with different output functions are less restrictive. More specifically, the algebraic criteria of the generalized model are applicable to several well-known neural network models by taking special parameters, and multiple different output functions are introduced to replace some of the same output functions, which improves the diversity of output results for the design of neural networks. In addition, the estimation of relative convergence rate of ψ -type stability is determined by the parameters of CGNNs and the selection of ψ -type functions. As a result, the existing results on multistability and monostability can be improved and extended. Finally, some numerical simulations are presented to illustrate the effectiveness of the obtained results.
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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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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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Zhang F, Zeng Z. Multistability and instability analysis of recurrent neural networks with time-varying delays. Neural Netw 2017; 97:116-126. [PMID: 29096200 DOI: 10.1016/j.neunet.2017.09.013] [Citation(s) in RCA: 30] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/11/2017] [Revised: 08/07/2017] [Accepted: 09/26/2017] [Indexed: 11/25/2022]
Abstract
This paper provides new theoretical results on the multistability and instability analysis of recurrent neural networks with time-varying delays. It is shown that such n-neuronal recurrent neural networks have exactly [Formula: see text] equilibria, [Formula: see text] of which are locally exponentially stable and the others are unstable, where k0 is a nonnegative integer such that k0≤n. By using the combination method of two different divisions, recurrent neural networks can possess more dynamic properties. This method improves and extends the existing results in the literature. Finally, one numerical example is provided to show the superiority and effectiveness of the presented results.
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Affiliation(s)
- Fanghai Zhang
- School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China; Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China.
| | - Zhigang Zeng
- School of Automation, Huazhong University of Science and Technology, Wuhan 430074, China; Key Laboratory of Image Processing and Intelligent Control of Education Ministry of China, Wuhan 430074, China.
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