Document Zbl 1525.93304

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The coexistence as well as the stability of multiple equilibrium points (EPs) of DRNNs are proved. With the Brouwer’s fixed point theorem as well as the Lagrange mean value theorem, it is obtained that under some conditions, the $n$-neuron DRNNs with the proposed activation function can have at least $5^n$ EPs and $3^n$ of them are locally stable. Compared with the DRNNs with sigmoidal activation functions, DRNNs with this kind of activation function can have more total EPs and more locally stable EPs. It implies that when designing DRNNs with the proposed activation function to apply in associative memory, it can have an even larger storage capacity. Furthermore, it is obtained that there exists a relationship between the number of the total EPs/stable EPs and the frequency of the sinusoidal function in the proposed activation function. Last, the above obtained results are extended to a more general case. It is shown that, DRNNs with the extended activation function can have $(2k+1)^n$ EPs, $(k+1)^n$ of which are locally stable, therein $k$ is closely related to the frequency of the sinusoidal function in the extended activation function. Two simulation examples are given to verify the correctness of the theoretical results.</div> <div class="clear"></div> <br> <div class="citations"><div class="clear"><a href="/?q=rf%3A7751338">Cited in <strong>5</strong> Documents</a></div></div> <div class="classification"> <h3>MSC:</h3> <table><tr> <td> <a class="mono" href="/classification/?q=cc%3A93D05" title="MSC2020">93D05</a> </td> <td class="space"> Lyapunov and other classical stabilities (Lagrange, Poisson, $L^p, l^p$, etc.) in control theory </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93B70" title="MSC2020">93B70</a> </td> <td class="space"> Networked control </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C43" title="MSC2020">93C43</a> </td> <td class="space"> Delay control/observation systems </td> </tr><tr> <td> <a class="mono" href="/classification/?q=cc%3A93C10" title="MSC2020">93C10</a> </td> <td class="space"> Nonlinear systems in control theory </td> </tr></table> </div><div class="keywords"> <h3>Keywords:</h3><a href="/?q=ut%3Amultistability">multistability</a>; <a href="/?q=ut%3Arecurrent+neural+networks">recurrent neural networks</a>; <a href="/?q=ut%3Atime-varying+delay">time-varying delay</a>; <a href="/?q=ut%3Apiecewise+nonlinear+activation+functions">piecewise nonlinear activation functions</a>; <a href="/?q=ut%3Aequilibrium+points">equilibrium points</a></div>  <div class="modal fade" id="metadataModal" tabindex="-1" role="dialog" aria-labelledby="myModalLabel"> <div class="modal-dialog" role="document"> <div class="modal-content"> <div class="modal-header"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> <h4 class="modal-title" id="myModalLabel">Cite</h4> </div> <div class="modal-body"> <div class="form-group"> <label for="select-output" class="control-label">Format</label> <select id="select-output" class="form-control" aria-label="Select Metadata format"></select> </div> <div class="form-group"> <label for="metadataText" class="control-label">Result</label> <textarea class="form-control" id="metadataText" rows="10" style="min-width: 100%;max-width: 100%"></textarea> </div> <div id="metadata-alert" class="alert alert-danger" role="alert" style="display: none;">  </div> </div> <div class="modal-footer"> <button type="button" class="btn btn-primary" onclick="copyMetadata()">Copy to clipboard</button> <button type="button" class="btn btn-default" data-dismiss="modal">Close</button> </div> </div> </div> </div> <div class="functions clearfix"> <div class="function">  <a type="button" class="btn btn-default btn-xs pdf" data-toggle="modal" data-target="#metadataModal" data-itemtype="Zbl" data-itemname="Zbl 1525.93304" data-ciurl="/ci/07751338" data-biburl="/bibtex/07751338.bib" data-amsurl="/amsrefs/07751338.bib" data-xmlurl="/xml/07751338.xml" > Cite </a> <a class="btn btn-default btn-xs pdf" data-container="body" type="button" href="/pdf/07751338.pdf" title="Zbl 1525.93304 as PDF">Review PDF</a> </div> <div class="fulltexts"> <span class="fulltext">Full Text:</span> <a class="btn btn-default btn-xs" type="button" href="https://doi.org/10.1016/j.neunet.2022.04.015" aria-label="DOI for “Multistability analysis of delayed recurrent neural networks with a class of piecewise nonlinear activation functions”" title="10.1016/j.neunet.2022.04.015">DOI</a> </div> <div class="sfx" style="float: right;"> </div> </div> <div class="references"> <h3>References:</h3> <table><tr> <td>[1]</td> <td class="space">Bao, H.; Park, J. H.; Cao, J., Non-fragile state estimation for fractional-order delayed memristive BAM neural networks, Neural Networks, 119, 190-199 (2019) · <a href="/1443.93107" class="nowrap">Zbl 1443.93107</a></td> </tr><tr> <td>[2]</td> <td class="space">Bao, H.; Park, J. H.; Cao, J., Adaptive synchronization of fractional-order output-coupling neural networks via quantized output control, IEEE Transactions on Neural Networks and Learning Systems, 32, 7, 3230-3239 (2020)</td> </tr><tr> <td>[3]</td> <td class="space">Chen, W. H.; Luo, S.; Lu, X., Multistability in a class of stochastic delayed hopfield neural networks, Neural Networks, 68, 52-61 (2015) · <a href="/1398.34119" class="nowrap">Zbl 1398.34119</a></td> </tr><tr> <td>[4]</td> <td class="space">Cheng, C. Y.; Lin, K. H.; Shih, C. W., Multistability in recurrent neural networks, SIAM Journal of Applied Mathematics, 66, 4, 1301-1320 (2006) · <a href="/1106.34048" class="nowrap">Zbl 1106.34048</a></td> </tr><tr> <td>[5]</td> <td class="space">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 (2021)</td> </tr><tr> <td>[6]</td> <td class="space">Ensari, T.; Arik, S., Global stability analysis of neural networks with multiple time varying delays, IEEE Transactions on Automatic Control, 50, 11, 1781-1785 (2005) · <a href="/1365.93423" class="nowrap">Zbl 1365.93423</a></td> </tr><tr> <td>[7]</td> <td class="space">Forti, M.; Tesi, A., New conditions for global stability of neural networks with application to linear and quadratic programming problems, IEEE Transactions on Circuits and Systems I, 42, 7, 354-366 (1995) · <a href="/0849.68105" class="nowrap">Zbl 0849.68105</a></td> </tr><tr> <td>[8]</td> <td class="space">Guo, Z.; Liu, L.; Wang, J., Multistability for recurrent neural networks with piecewise-linear radial basis functions and state-dependent switching parameters, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 50, 11, 4458-4471 (2018)</td> </tr><tr> <td>[9]</td> <td class="space">Guo, Z.; Liu, L.; Wang, J., Multistability of switched neural networks with piecewise linear activation functions under state-dependent switching, IEEE Transactions on Neural Networks and Learning Systems, 30, 7, 2052-2066 (2018)</td> </tr><tr> <td>[10]</td> <td class="space">Guo, Z.; Ou, S.; Wang, J., Multistability of switched neural networks with sigmoidal activation functions under state-dependent switching, Neural Networks, 122, 239-252 (2020)</td> </tr><tr> <td>[11]</td> <td class="space">Hu, J.; Wang, J., Global stability of complex-valued recurrent neural networks with time-delays, IEEE Transactions on Neural Networks and Learning Systems, 23, 6, 853-865 (2012)</td> </tr><tr> <td>[12]</td> <td class="space">Huang, G.; Cao, J., Multistability in bidirectional associative memory neural networks, Physics Letters. A, 372, 16, 2842-2854 (2008) · <a href="/1220.92002" class="nowrap">Zbl 1220.92002</a></td> </tr><tr> <td>[13]</td> <td class="space">Huang, G.; Cao, J., Multistability of neural networks with discontinuous activation function, Communications in Nonlinear Science and Numerical Simulation, 13, 10, 2279-2289 (2008) · <a href="/1221.34131" class="nowrap">Zbl 1221.34131</a></td> </tr><tr> <td>[14]</td> <td class="space">Huang, Z.; Mohamad, S.; Cai, G., $ 2^N$ Almost periodic attractors for CNNs with variable and distributed delays, Journal of the Franklin Institute, 346, 4, 391-412 (2009) · <a href="/1176.34086" class="nowrap">Zbl 1176.34086</a></td> </tr><tr> <td>[15]</td> <td class="space">Huang, Z.; Mohamad, S.; Feng, C.; Cai, G., Coexistence of $2^N$ domains of attraction of nonautonomous neural networks with time-varying delays, Applied Mathematical Modelling, 34, 1, 80-91 (2010) · <a href="/1185.34073" class="nowrap">Zbl 1185.34073</a></td> </tr><tr> <td>[16]</td> <td class="space">Huang, Z.; Wang, X.; Feng, C., Multiperiodicity of periodically oscillated discrete-time neural networks with transient excitatory self-connections and sigmoidal nonlinearities, IEEE Transactions on Neural Networks, 21, 10, 1643-1655 (2010)</td> </tr><tr> <td>[17]</td> <td class="space">Huang, Y.; Zhang, X., Multistability properties of linear threshold discrete-time recurrent neural networks, International Journal of Information & Systems Sciences, 7, 1, 1-10 (2011) · <a href="/1498.39016" class="nowrap">Zbl 1498.39016</a></td> </tr><tr> <td>[18]</td> <td class="space">Huang, Y.; Zhang, H.; Wang, Z., Multistability and multiperiodicity of delayed bidirectional associative memory neural networks with discontinuous activation functions, Applied Mathematics and Computation, 219, 3, 899-910 (2012) · <a href="/1291.34115" class="nowrap">Zbl 1291.34115</a></td> </tr><tr> <td>[19]</td> <td class="space">Kaslik, E.; Sivasundaram, S., Multiple periodic solutions in impulsive hybrid neural networks with delays, Applied Mathematics and Computation, 217, 10, 4890-4899 (2011) · <a href="/1225.34071" class="nowrap">Zbl 1225.34071</a></td> </tr><tr> <td>[20]</td> <td class="space">Kaslik, E.; Sivasundaram, S., Multistability in impulsive hybrid hopfield neural networks with distributed delays, Nonlinear Analysis. Real World Applications, 12, 3, 1640-1649 (2011) · <a href="/1221.34219" class="nowrap">Zbl 1221.34219</a></td> </tr><tr> <td>[21]</td> <td class="space">Liu, Y.; Huang, X.; Li, Y.; Shen, H., Multistability of hopfield neural networks with a designed discontinuous sawtooth-type activation function, Neurocomputing, 455, 189-201 (2021)</td> </tr><tr> <td>[22]</td> <td class="space">Liu, P.; Nie, X.; Liang, J.; Cao, J., Multiple Mittag-Leffler stability of fractional-order competitive neural networks with Gaussian activation functions, Neural Networks, 108, 452-465 (2018) · <a href="/1441.93282" class="nowrap">Zbl 1441.93282</a></td> </tr><tr> <td>[23]</td> <td class="space">Liu, Y.; Wu, J.; Wang, X., Collective periodic motions in a multiparticle model involving processing delay, Mathematical Methods in the Applied Sciences, 44, 5, 3280-3302 (2021) · <a href="/1478.34091" class="nowrap">Zbl 1478.34091</a></td> </tr><tr> <td>[24]</td> <td class="space">Liu, P.; Zeng, Z.; Wang, J., Multistability of recurrent neural networks with nonmonotonic activation functions and mixed time delays, IEEE Transactions on Systems Man and Cybernetics-Systems, 46, 4, 512-523 (2016)</td> </tr><tr> <td>[25]</td> <td class="space">Long, X., Novel stability criteria on a patch structure nicholson’s blowflies model with multiple pairs of time-varying delays, AIMS Mathematics, 5, 6, 7387-7401 (2020) · <a href="/1484.34151" class="nowrap">Zbl 1484.34151</a></td> </tr><tr> <td>[26]</td> <td class="space">Lu, W.; Chen, T., Dynamical behaviors of cohen-grossberg neural networks with discontinuous activation functions, Neural Networks, 18, 3, 231-242 (2005) · <a href="/1078.68127" class="nowrap">Zbl 1078.68127</a></td> </tr><tr> <td>[27]</td> <td class="space">Manickam, I.; Ramachandran, R.; Rajchakit, G.; Cao, J.; Huang, C., Novel Lagrange sense exponential stability criteria for time-delayed stochastic cohen-grossberg neural networks with Markovian jump parameters: A graph-theoretic approach, Nonlinear Analysis: Modelling and Control, 25, 5, 726-744 (2020) · <a href="/1448.93273" class="nowrap">Zbl 1448.93273</a></td> </tr><tr> <td>[28]</td> <td class="space">Morita, M., Associative memory with nonmonotone dynamics, Neural Networks, 6, 1, 115-126 (1993)</td> </tr><tr> <td>[29]</td> <td class="space">Murino, V., Structured neural networks for pattern recognition, IEEE Transactions on Systems, Man and Cybernetics, Part B, 28, 4, 553-561 (1998)</td> </tr><tr> <td>[30]</td> <td class="space">Nie, X.; Cao, J., Multistability of competitive neural networks with time-varying and distributed delays, Nonlinear Analysis. Real World Applications, 10, 2, 928-942 (2009) · <a href="/1167.34383" class="nowrap">Zbl 1167.34383</a></td> </tr><tr> <td>[31]</td> <td class="space">Nie, X.; Cao, J.; Fei, S., Multistability and instability of competitive neural networks with non-monotonic piecewise linear activation functions, Nonlinear Analysis. Real World Applications, 45, 799-821 (2019) · <a href="/1417.34118" class="nowrap">Zbl 1417.34118</a></td> </tr><tr> <td>[32]</td> <td class="space">Nie, X.; Liang, J.; Cao, J., Multistability analysis of competitive neural networks with Gaussian-wavelet-type activation functions and unbounded time-varying delays, Applied Mathematics and Computation, 356, 449-468 (2019) · <a href="/1428.34105" class="nowrap">Zbl 1428.34105</a></td> </tr><tr> <td>[33]</td> <td class="space">Nie, X.; Zheng, W. X., Dynamical behaviors of multiple equilibria in competitive neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Transactions on Cybernetics, 46, 3, 679-693 (2015)</td> </tr><tr> <td>[34]</td> <td class="space">Nie, X.; Zheng, W. X., Multistability and instability of neural networks with discontinuous nonmonotonic piecewise linear activation functions, IEEE Transactions on Neural Networks and Learning Systems, 26, 11, 2901-2913 (2015)</td> </tr><tr> <td>[35]</td> <td class="space">Qi, X.; Bao, H.; Cao, J., Synchronization criteria for quaternion-valued coupled neural networks with impulses, Neural Networks, 128, 150-157 (2020) · <a href="/1471.93225" class="nowrap">Zbl 1471.93225</a></td> </tr><tr> <td>[36]</td> <td class="space">Shen, Y.; Zhu, S.; Liu, X.; Wen, S., Multistability and associative memory of neural networks with morita-like activation functions, Neural Networks, 142, 162-170 (2021) · <a href="/1536.92010" class="nowrap">Zbl 1536.92010</a></td> </tr><tr> <td>[37]</td> <td class="space">Tu, Z.; Jian, J.; Wang, B., Positive invariant sets and global exponential attractive sets of a class of neural networks with unbounded time-delays, Communications in Nonlinear Science and Numerical Simulation, 16, 9, 3738-3745 (2011) · <a href="/1252.34085" class="nowrap">Zbl 1252.34085</a></td> </tr><tr> <td>[38]</td> <td class="space">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 Networks, 122, 231-238 (2020)</td> </tr><tr> <td>[39]</td> <td class="space">Wang, L.; Chen, T., Multistability of neural networks with mexican-hat-type activation functions, IEEE Transactions on Neural Networks and Learning Systems, 23, 11, 1816-1826 (2012)</td> </tr><tr> <td>[40]</td> <td class="space">Wang, Q.; Fang, Y.; Li, H.; Su, L.; Dai, B., Anti-periodic solutions for high-order hopfield neural networks with impulses, Neurocomputing, 138, 339-346 (2014)</td> </tr><tr> <td>[41]</td> <td class="space">Wang, J.; He, S.; Huang, L., Limit cycles induced by threshold nonlinearity in planar piecewise linear systems of node-focus or node-center type, International Journal of Bifurcation and Chaos, 30, 11, Article 2050160 pp. (2020) · <a href="/1478.34037" class="nowrap">Zbl 1478.34037</a></td> </tr><tr> <td>[42]</td> <td class="space">Yi, Z.; Tan, K. K., Multistability of discrete-time recurrent neural networks with unsaturating piecewise linear activation functions, IEEE Transactions on Neural Networks, 15, 2, 329-336 (2004), Networks, 21(10), 1643-1655</td> </tr><tr> <td>[43]</td> <td class="space">Yi, Z.; Tan, K. K.; Lee, T. H., Multistability analysis for recurrent neural networks with unsaturating piecewise linear transfer functions, Neural Computation, 15, 3, 639-662 (2003) · <a href="/1085.68142" class="nowrap">Zbl 1085.68142</a></td> </tr><tr> <td>[44]</td> <td class="space">Zeng, Z.; Huang, D. S.; Wang, Z., Memory pattern analysis of cellular neural networks, Physics Letters. A, 342, 1-2, 114-128 (2005) · <a href="/1222.92013" class="nowrap">Zbl 1222.92013</a></td> </tr><tr> <td>[45]</td> <td class="space">Zeng, Z.; Huang, T.; Zheng, W. X., Multistability of recurrent neural networks with time-varying delays and the piecewise linear activation function, IEEE Transactions on Neural Networks, 21, 8, 1371-1377 (2010)</td> </tr><tr> <td>[46]</td> <td class="space">Zeng, Z.; Wang, J., Multiperiodicity of discrete-time delayed neural networks evoked by periodic external inputs, IEEE Transactions on Neural Networks, 17, 5, 1141-1151 (2006)</td> </tr><tr> <td>[47]</td> <td class="space">Zeng, Z.; Wang, J.; Liao, X., Stability analysis of delayed cellular neural networks described using cloning templates, IEEE Transactions on Circuits and Systems. I. Regular Papers, 51, 11, 2313-2324 (2004) · <a href="/1374.34293" class="nowrap">Zbl 1374.34293</a></td> </tr><tr> <td>[48]</td> <td class="space">Zhang, J.; Huang, C., Dynamics analysis on a class of delayed neural networks involving inertial terms, Advances in Difference Equations, 2020, 1, 1-12 (2020) · <a href="/1482.34177" class="nowrap">Zbl 1482.34177</a></td> </tr><tr> <td>[49]</td> <td class="space">Zhang, H.; Huang, Y.; Wang, B.; Wang, Z., Design and analysis of associative memories based on external inputs of delayed recurrent neural networks, Neurocomputing, 136, 337-344 (2014)</td> </tr></table> <div class="reference_disclaimer"> This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. 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