Bifurcation in a nonlinear steady state system
The steady state solutions of a nonlinear digital cellular neural network with \(\omega\) neural units and a nonnegative variable parameter \(\lambda\) are sought. We show that \(\lambda = 1\) is a critical value such that the qualitative behavior of our network changes. More specifically, when \(\o...
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| Format: | Article |
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AGH Univeristy of Science and Technology Press
2010-01-01
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| Series: | Opuscula Mathematica |
| Subjects: | |
| Online Access: | http://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3027.pdf |
| _version_ | 1818619818925883392 |
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| author | Gen-Qiang Wang Sui Sun Cheng |
| author_facet | Gen-Qiang Wang Sui Sun Cheng |
| author_sort | Gen-Qiang Wang |
| collection | DOAJ |
| description | The steady state solutions of a nonlinear digital cellular neural network with \(\omega\) neural units and a nonnegative variable parameter \(\lambda\) are sought. We show that \(\lambda = 1\) is a critical value such that the qualitative behavior of our network changes. More specifically, when \(\omega\) is odd, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda \in [1,\infty)\), steady states cannot exist; while when \(\omega\) is even, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda = 1\), there are no nontrivial steady states, and for \(\lambda \in (1,\infty)\), there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers. |
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| format | Article |
| id | doaj.art-e0e0b5334b1845d3b63f79a6492037d8 |
| institution | Directory Open Access Journal |
| issn | 1232-9274 |
| language | English |
| last_indexed | 2024-12-16T17:43:32Z |
| publishDate | 2010-01-01 |
| publisher | AGH Univeristy of Science and Technology Press |
| record_format | Article |
| series | Opuscula Mathematica |
| spelling | doaj.art-e0e0b5334b1845d3b63f79a6492037d82022-12-21T22:22:32ZengAGH Univeristy of Science and Technology PressOpuscula Mathematica1232-92742010-01-01303349360http://dx.doi.org/10.7494/OpMath.2010.30.3.3493027Bifurcation in a nonlinear steady state systemGen-Qiang Wang0Sui Sun Cheng1Guangdone Polytechnic Normal University, Department of Computer Science, Guangzhou, Guangdone 510665, P. R. ChinaTsing Hua University, Department of Mathematics, Hsinchu, Taiwan 30043, R. O. ChinaThe steady state solutions of a nonlinear digital cellular neural network with \(\omega\) neural units and a nonnegative variable parameter \(\lambda\) are sought. We show that \(\lambda = 1\) is a critical value such that the qualitative behavior of our network changes. More specifically, when \(\omega\) is odd, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda \in [1,\infty)\), steady states cannot exist; while when \(\omega\) is even, then for \(\lambda \in [0,1)\), there is one positive and one negative steady state, and for \(\lambda = 1\), there are no nontrivial steady states, and for \(\lambda \in (1,\infty)\), there are two fully oscillatory steady states. Furthermore, the number of existing nontrivial solutions cannot be improved. It is hoped that our results are of interest to digital neural network designers.http://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3027.pdfbifurcationcellular neural networksteady stateKrasnoselsky fixed point theorem |
| spellingShingle | Gen-Qiang Wang Sui Sun Cheng Bifurcation in a nonlinear steady state system Opuscula Mathematica bifurcation cellular neural network steady state Krasnoselsky fixed point theorem |
| title | Bifurcation in a nonlinear steady state system |
| title_full | Bifurcation in a nonlinear steady state system |
| title_fullStr | Bifurcation in a nonlinear steady state system |
| title_full_unstemmed | Bifurcation in a nonlinear steady state system |
| title_short | Bifurcation in a nonlinear steady state system |
| title_sort | bifurcation in a nonlinear steady state system |
| topic | bifurcation cellular neural network steady state Krasnoselsky fixed point theorem |
| url | http://www.opuscula.agh.edu.pl/vol30/3/art/opuscula_math_3027.pdf |
| work_keys_str_mv | AT genqiangwang bifurcationinanonlinearsteadystatesystem AT suisuncheng bifurcationinanonlinearsteadystatesystem |