Multistability of complex-valued neural networks with discontinuous activation functions

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• 作者：Jinling Liang^a ; ^{jinlliang@seu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Weiqiang Gong^a ; ^{gong_wei_qiang@163.com" class="auth_mail" title="E-mail the corresponding author} ; Tingwen Huang^b ; ^{tingwen.huang@qatar.tamu.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Complex-valued neural networks ; Multistability ; Discontinuous function ; Attraction basin
• 刊名：Neural Networks
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：84
• 期：Complete
• 页码：125-142
• 全文大小：2841 K

文摘

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 class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893608016301125&_mathId=si19.gif&_user=111111111&_pii=S0893608016301125&_rdoc=1&_issn=08936080&md5=72e16a46dcf8dd4e320f35309d664fc5" title="Click to view the MathML source">25ⁿclass="mathContainer hidden">class="mathCode">$25^n$ equilibrium points. Among them, class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893608016301125&_mathId=si20.gif&_user=111111111&_pii=S0893608016301125&_rdoc=1&_issn=08936080&md5=372c94abcd6ae18f86ac675765d93241" title="Click to view the MathML source">9ⁿclass="mathContainer hidden">class="mathCode">$9^n$ points are locally stable and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893608016301125&_mathId=si21.gif&_user=111111111&_pii=S0893608016301125&_rdoc=1&_issn=08936080&md5=ec135be54a9cf8858b77a3c5211f74a6" title="Click to view the MathML source">16ⁿ−9ⁿclass="mathContainer hidden">class="mathCode">$16^n-9^n$ equilibrium points are unstable. Furthermore, to enlarge the attraction basins of the class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893608016301125&_mathId=si20.gif&_user=111111111&_pii=S0893608016301125&_rdoc=1&_issn=08936080&md5=372c94abcd6ae18f86ac675765d93241" title="Click to view the MathML source">9ⁿclass="mathContainer hidden">class="mathCode">$9^n$ equilibrium points, some mild conditions are imposed. Finally, one numerical example is provided to illustrate the effectiveness of the obtained results.