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Kong X, Yu F, Yao W, Cai S, Zhang J, Lin H. Memristor-induced hyperchaos, multiscroll and extreme multistability in fractional-order HNN: Image encryption and FPGA implementation. Neural Netw 2024; 171:85-103. [PMID: 38091767 DOI: 10.1016/j.neunet.2023.12.008] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/21/2023] [Revised: 11/06/2023] [Accepted: 12/04/2023] [Indexed: 01/29/2024]
Abstract
Fractional-order differentiation (FOD) can record information from the past, present, and future. Compared with integer-order systems, FOD systems have higher complexity and more accurate ability to describe the real world. In this paper, two types of fractional-order memristors are proposed and one type is proved to have extreme multistability, local activity, and non-volatility. By using memristors to simulate the autapse of a neuron and to describe the phenomenon of electromagnetic induction caused by electromagnetic radiation, we establish a new 5D FOD memristive HNN (FOMHNN). Through dynamic simulation, rich dynamic behaviors are found, such as hyperchaos, multiscroll, extreme multistability, and "overclocking" behavior caused by order reduction. To the best of our knowledge, this is the first time that such rich dynamic behaviors are found in FOMHNN simultaneously. Based on this FOMHNN, a very efficient and secure image encryption scheme is designed. Security analysis shows that the encrypted Lena image has extremely low adjacent pixel correlation and high randomness, with information entropy of 7.9995. Despite discarding diffusion and scrambling, it has excellent plaintext sensitivity, with NCPR = 99.6095% and UACI = 33.4671%. Finally, this paper implements the proposed FOMHNN and image encryption on field programmable gate array (FPGA). To our knowledge, the related work of fully hardware implementation of fractional-order neural networks and image encryption schemes based on this is rare.
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Affiliation(s)
- Xinxin Kong
- School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, China.
| | - Fei Yu
- School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, China.
| | - Wei Yao
- School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, China.
| | - Shuo Cai
- School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, China.
| | - Jin Zhang
- School of Computer and Communication Engineering, Changsha University of Science and Technology, Changsha, 410114, Hunan, China.
| | - Hairong Lin
- School of Computer and Communication Engineering, Hunan University, Changsha, 410082, Hunan, China.
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2
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Wu Z, Nie X, Cao B. Coexistence and local stability of multiple equilibrium points for fractional-order state-dependent switched competitive neural networks with time-varying delays. Neural Netw 2023; 160:132-147. [PMID: 36640489 DOI: 10.1016/j.neunet.2022.12.013] [Citation(s) in RCA: 2] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/04/2022] [Revised: 11/09/2022] [Accepted: 12/16/2022] [Indexed: 01/05/2023]
Abstract
This paper investigates the coexistence and local stability of multiple equilibrium points for a class of competitive neural networks with sigmoidal activation functions and time-varying delays, in which fractional-order derivative and state-dependent switching are involved at the same time. Some novel criteria are established to ensure that such n-neuron neural networks can have [Formula: see text] total equilibrium points and [Formula: see text] locally stable equilibrium points with m1+m2=n, based on the fixed-point theorem, the definition of equilibrium point in the sense of Filippov, the theory of fractional-order differential equation and Lyapunov function method. The investigation implies that the competitive neural networks with switching can possess greater storage capacity than the ones without switching. Moreover, the obtained results include the multistability results of both fractional-order switched Hopfield neural networks and integer-order switched Hopfield neural networks as special cases, thus generalizing and improving some existing works. Finally, four numerical examples are presented to substantiate the effectiveness of the theoretical analysis.
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Affiliation(s)
- Zhongwen Wu
- School of Mathematics, Southeast University, Nanjing, 211189, China.
| | - Xiaobing Nie
- School of Mathematics, Southeast University, Nanjing, 211189, China.
| | - Boqiang Cao
- School of Mathematics, Southeast University, Nanjing, 211189, China.
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3
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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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4
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Deng K, Zhu S, Bao G, Fu J, Zeng Z. Multistability of Dynamic Memristor Delayed Cellular Neural Networks With Application to Associative Memories. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2023; 34:690-702. [PMID: 34347606 DOI: 10.1109/tnnls.2021.3099814] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
Abstract
Recently, dynamic memristor (DM)-cellular neural networks (CNNs) have received widespread attention due to their advantage of low power consumption. The previous works showed that DM-CNNs have at most 318 equilibrium points (EPs) with n=16 cells. Since time delay is unavoidable during the process of information transmission, the goal of this article is to research the multistability of DM-CNNs with time delay, and, meanwhile, to increase the storage capacity of DM-delay (D)CNNs. Depending on the different constitutive relations of memristors, two cases of the multistability for DM-DCNNs are discussed. After determining the constitutive relations, the number of EPs of DM-DCNNs is increased to 3n with n cells by means of the appropriate state-space decomposition and the Brouwer's fixed point theorem. Furthermore, the enlarged attraction domains of EPs can be obtained, and 2n of these EPs are locally exponentially stable in two cases. Compared with standard CNNs, the dynamic behavior of DM-DCNNs shows an outstanding merit. That is, the value of voltage and current approach to zero when the system becomes stable, and the memristor provides a nonvolatile memory to store the computation results. Finally, two numerical simulations are presented to illustrate the effectiveness of the theoretical results, and the applications of associative memories are shown at the end of this article.
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5
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Chen L, Gu P, Lopes AM, Chai Y, Xu S, Ge S. Asymptotic Stability of Fractional-Order Incommensurate Neural Networks. Neural Process Lett 2022. [DOI: 10.1007/s11063-022-11095-y] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/03/2022]
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6
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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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7
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Multistability analysis of switched fractional-order recurrent neural networks with time-varying delay. Neural Comput Appl 2022. [DOI: 10.1007/s00521-022-07592-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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8
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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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Solís-Pérez J, Hernández J, Parrales A, Gómez-Aguilar J, Huicochea A. Artificial neural networks with conformable transfer function for improving the performance in thermal and environmental processes. Neural Netw 2022; 152:44-56. [DOI: 10.1016/j.neunet.2022.04.016] [Citation(s) in RCA: 2] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2021] [Revised: 03/19/2022] [Accepted: 04/14/2022] [Indexed: 10/18/2022]
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10
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Multistability analysis of delayed recurrent neural networks with a class of piecewise nonlinear activation functions. Neural Netw 2022; 152:80-89. [DOI: 10.1016/j.neunet.2022.04.015] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/10/2021] [Revised: 04/11/2022] [Accepted: 04/14/2022] [Indexed: 11/15/2022]
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11
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Saravanakumar R, Ali MS. Extended Dissipative Criteria for Generalized Markovian Jump Neural Networks Including Asynchronous Mode-Dependent Delayed States. Neural Process Lett 2022. [DOI: 10.1007/s11063-021-10697-2] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/27/2022]
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12
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Improved Results on Finite-Time Passivity and Synchronization Problem for Fractional-Order Memristor-Based Competitive Neural Networks: Interval Matrix Approach. FRACTAL AND FRACTIONAL 2022. [DOI: 10.3390/fractalfract6010036] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This research paper deals with the passivity and synchronization problem of fractional-order memristor-based competitive neural networks (FOMBCNNs) for the first time. Since the FOMBCNNs’ parameters are state-dependent, FOMBCNNs may exhibit unexpected parameter mismatch when different initial conditions are chosen. Therefore, the conventional robust control scheme cannot guarantee the synchronization of FOMBCNNs. Under the framework of the Filippov solution, the drive and response FOMBCNNs are first transformed into systems with interval parameters. Then, the new sufficient criteria are obtained by linear matrix inequalities (LMIs) to ensure the passivity in finite-time criteria for FOMBCNNs with mismatched switching jumps. Further, a feedback control law is designed to ensure the finite-time synchronization of FOMBCNNs. Finally, three numerical cases are given to illustrate the usefulness of our passivity and synchronization results.
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13
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Xiang J, Tan M. Dynamic behavior analysis of Stepanov-like almost periodic solution in distribution sense for stochastic neural network with delays. Neurocomputing 2022. [DOI: 10.1016/j.neucom.2021.10.108] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/19/2022]
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14
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Synchronization of Fractional Order Uncertain BAM Competitive Neural Networks. FRACTAL AND FRACTIONAL 2021. [DOI: 10.3390/fractalfract6010014] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
This article examines the drive-response synchronization of a class of fractional order uncertain BAM (Bidirectional Associative Memory) competitive neural networks. By using the differential inclusions theory, and constructing a proper Lyapunov-Krasovskii functional, novel sufficient conditions are obtained to achieve global asymptotic stability of fractional order uncertain BAM competitive neural networks. This novel approach is based on the linear matrix inequality (LMI) technique and the derived conditions are easy to verify via the LMI toolbox. Moreover, numerical examples are presented to show the feasibility and effectiveness of the theoretical results.
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15
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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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16
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Multistability of Hopfield neural networks with a designed discontinuous sawtooth-type activation function. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.05.045] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/23/2022]
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17
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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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18
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Xu Y, Gao S, Li W. Exponential Stability of Fractional-Order Complex Multi-Links Networks With Aperiodically Intermittent Control. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2021; 32:4063-4074. [PMID: 32894724 DOI: 10.1109/tnnls.2020.3016672] [Citation(s) in RCA: 17] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
In this article, the exponential stability problem for fractional-order complex multi-links networks with aperiodically intermittent control is considered. Using the graph theory and Lyapunov method, two theorems, including a Lyapunov-type theorem and a coefficient-type theorem, are given to ensure the exponential stability of the underlying networks. The theoretical results show that the exponential convergence rate is dependent on the control gain and the order of fractional derivative. To be specific, the larger control gain, the higher the exponential convergence rate. Meanwhile, when aperiodically intermittent control degenerates into periodically intermittent control, a corollary is also provided to ensure the exponential stability of the underlying networks. Furthermore, to show the practicality of theoretical results, as an application, the exponential stability of fractional-order multi-links competitive neural networks with aperiodically intermittent control is investigated and a stability criterion is established. Finally, the effectiveness and feasibility of the theoretical results are demonstrated through a numerical example.
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Nie X, Liu P, Liang J, Cao J. Exact coexistence and locally asymptotic stability of multiple equilibria for fractional-order delayed Hopfield neural networks with Gaussian activation function. Neural Netw 2021; 142:690-700. [PMID: 34403909 DOI: 10.1016/j.neunet.2021.07.029] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/22/2021] [Revised: 06/08/2021] [Accepted: 07/26/2021] [Indexed: 11/30/2022]
Abstract
This paper explores the multistability issue for fractional-order Hopfield neural networks with Gaussian activation function and multiple time delays. First, several sufficient criteria are presented for ensuring the exact coexistence of 3n equilibria, based on the geometric characteristics of Gaussian function, the fixed point theorem and the contraction mapping principle. Then, different from the existing methods used in the multistability analysis of fractional-order neural networks without time delays, it is shown that 2n of 3n total equilibria are locally asymptotically stable, by applying the theory of fractional-order linear delayed system and constructing suitable Lyapunov function. The obtained results improve and extend some existing multistability works for classical integer-order neural networks and fractional-order neural networks without time delays. Finally, an illustrative example with comprehensive computer simulations is given to demonstrate the theoretical results.
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Affiliation(s)
- Xiaobing Nie
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China.
| | - Pingping Liu
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China
| | - Jinling Liang
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China
| | - Jinde Cao
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China; Yonsei Frontier Lab, Yonsei University, Seoul 03722, South Korea
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20
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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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21
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Exponential stabilization for fractional intermittent controlled multi-group models with dispersal. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.02.063] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
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22
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Cao B, Nie X. Event-triggered adaptive neural networks control for fractional-order nonstrict-feedback nonlinear systems with unmodeled dynamics and input saturation. Neural Netw 2021; 142:288-302. [PMID: 34082285 DOI: 10.1016/j.neunet.2021.05.014] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2020] [Revised: 04/10/2021] [Accepted: 05/10/2021] [Indexed: 10/21/2022]
Abstract
The event-triggered adaptive neural networks control is investigated in this paper for a class of fractional-order systems (FOSs) with unmodeled dynamics and input saturation. Firstly, in order to obtain an auxiliary signal and then avoid the state variables of unmodeled dynamics directly appearing in the designed controller, the notion of exponential input-to-state practical stability (ISpS) and some related lemmas for integer-order systems are extended to the ones for FOSs. Then, based on the traditional event-triggered mechanism, we propose a novel adaptive event-triggered mechanism (AETM) in this paper, in which the threshold parameters can be adjusted dynamically according to the tracking performance. Besides, different from the previous works where the derivative of hyperbolic tangent function tanh(⋅) needs to have positive lower bound, a new type of auxiliary signal is introduced in this paper to handle the effect of input saturation and thus this limitation is released. Finally, two numerical examples and some comparisons are provided to illustrate our proposed controllers.
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Affiliation(s)
- Boqiang Cao
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China.
| | - Xiaobing Nie
- The Jiangsu Provincial Key Laboratory of Networked Collective Intelligence, and School of Mathematics, Southeast University, Nanjing 211189, China.
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23
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Yang S, Jiang H, Hu C, Yu J. Synchronization for fractional-order reaction–diffusion competitive neural networks with leakage and discrete delays. Neurocomputing 2021. [DOI: 10.1016/j.neucom.2021.01.009] [Citation(s) in RCA: 13] [Impact Index Per Article: 4.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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24
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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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25
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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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26
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Wan P, Sun D, Zhao M, Zhao H. Monostability and Multistability for Almost-Periodic Solutions of Fractional-Order Neural Networks With Unsaturating Piecewise Linear Activation Functions. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 2020; 31:5138-5152. [PMID: 32092015 DOI: 10.1109/tnnls.2020.2964030] [Citation(s) in RCA: 6] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Since the unsaturating activation function is unbounded, more complex dynamics may exist in neural networks with this kind of activation function. In this article, monostability and multistability results of almost-periodic solutions are developed for fractional-order neural networks with unsaturating piecewise linear activation functions. Some globally Mittag-Leffler attractive sets are given, and the existence of globally Mittag-Leffler stable almost-periodic solution is demonstrated by using Ascoli-Arzela theorem. In particular, some sufficient conditions are provided to ascertain the multistability of almost-periodic solutions based on locally positively invariant set. It shows that there exists an almost-periodic solution in each positively invariant set, and all trajectories converge to this periodic trajectory in that rectangular area. Two illustrative examples are provided to demonstrate the effectiveness of the proposed sufficient criteria.
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27
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Liu XZ, Li ZT, Wu KN. Boundary Mittag-Leffler stabilization of fractional reaction-diffusion cellular neural networks. Neural Netw 2020; 132:269-280. [PMID: 32949988 DOI: 10.1016/j.neunet.2020.09.009] [Citation(s) in RCA: 14] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/01/2020] [Revised: 08/08/2020] [Accepted: 09/10/2020] [Indexed: 11/16/2022]
Abstract
Mittag-Leffler stabilization is studied for fractional reaction-diffusion cellular neural networks (FRDCNNs) in this paper. Different from previous literature, the FRDCNNs in this paper are high-dimensional systems, and boundary control and observed-based boundary control are both used to make FRDCNNs achieve Mittag-Leffler stability. First, a state-dependent boundary controller is designed when system states are available. By employing the spatial integral functional method and some inequalities, a criterion ensuring Mittag-Leffler stability of FRDCNNs is presented. Then, when the information of system states is not fully accessible, an observer is presented to estimate the system states based on boundary output and an observer-based boundary controller is provided aiming to stabilize the considered FRDCNNs. Furthermore, a robust observer-based boundary controller is proposed to ensure the Mittag-Leffler stability for FRDCNNs with uncertainties. Examples are given to illustrate the effectiveness of obtained theoretical results.
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Affiliation(s)
- Xiao-Zhen Liu
- Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China.
| | - Ze-Tao Li
- Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China.
| | - Kai-Ning Wu
- Department of Mathematics, Harbin Institute of Technology, Weihai, 264209, China.
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Yang S, Hu C, Yu J, Jiang H. Exponential Stability of Fractional-Order Impulsive Control Systems With Applications in Synchronization. IEEE TRANSACTIONS ON CYBERNETICS 2020; 50:3157-3168. [PMID: 30990206 DOI: 10.1109/tcyb.2019.2906497] [Citation(s) in RCA: 28] [Impact Index Per Article: 7.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/16/2023]
Abstract
This paper investigates exponential stability of fractional-order impulsive control systems (FICSs) and exponential synchronization of fractional-order Cohen-Grossberg neural networks (FCGNNs). First, under the framework of the generalized Caputo fractional-order derivative, some new results for fractional-order calculus are established by mainly using L'Hospital's rule and Laplace transform. Besides, FICSs are translated into impulsive differential equations with fractional-order via utilizing the definition of Dirac function, which reveals that the effect of impulsive control on fractional systems is dependent of the order of the addressed systems. Furthermore, exponential stability of FICSs is proposed and some novel criteria are obtained by applying average impulsive interval and the method of induction. As an application of the stability for FICSs, exponential synchronization of FCGNNs is considered and several synchronization conditions are established under impulsive control. Finally, several numerical examples are provided to illustrate the effectiveness of the derived results.
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29
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Global stability analysis of S-asymptotically ω-periodic oscillation in fractional-order cellular neural networks with time variable delays. Neurocomputing 2020. [DOI: 10.1016/j.neucom.2020.03.005] [Citation(s) in RCA: 13] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
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30
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He J, Chen F, Lei T, Bi Q. Global adaptive matrix-projective synchronization of delayed fractional-order competitive neural network with different time scales. Neural Comput Appl 2020. [DOI: 10.1007/s00521-020-04728-7] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/25/2022]
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31
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Finite-Time Mittag-Leffler Stability of Fractional-Order Quaternion-Valued Memristive Neural Networks with Impulses. Neural Process Lett 2019. [DOI: 10.1007/s11063-019-10154-1] [Citation(s) in RCA: 52] [Impact Index Per Article: 10.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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