Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays
A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by an...
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| Format: | Article |
| Language: | English |
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Wiley
2013-01-01
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| Series: | Journal of Applied Mathematics |
| Online Access: | http://dx.doi.org/10.1155/2013/436254 |
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| _version_ | 1832556122092863488 |
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| author | Zizhen Zhang Huizhong Yang |
| author_facet | Zizhen Zhang Huizhong Yang |
| author_sort | Zizhen Zhang |
| collection | DOAJ |
| description | A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis. |
| format | Article |
| id | doaj-art-b8d0837e75db4092836a1e7ec485805d |
| institution | Kabale University |
| issn | 1110-757X 1687-0042 |
| language | English |
| publishDate | 2013-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Journal of Applied Mathematics |
| spelling | doaj-art-b8d0837e75db4092836a1e7ec485805d2025-02-03T05:46:18ZengWileyJournal of Applied Mathematics1110-757X1687-00422013-01-01201310.1155/2013/436254436254Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two DelaysZizhen Zhang0Huizhong Yang1Key Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, ChinaKey Laboratory of Advanced Process Control for Light Industry (Ministry of Education), Jiangnan University, Wuxi 214122, ChinaA four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.http://dx.doi.org/10.1155/2013/436254 |
| spellingShingle | Zizhen Zhang Huizhong Yang Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays Journal of Applied Mathematics |
| title | Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays |
| title_full | Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays |
| title_fullStr | Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays |
| title_full_unstemmed | Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays |
| title_short | Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays |
| title_sort | hopf bifurcation analysis for a four dimensional recurrent neural network with two delays |
| url | http://dx.doi.org/10.1155/2013/436254 |
| work_keys_str_mv | AT zizhenzhang hopfbifurcationanalysisforafourdimensionalrecurrentneuralnetworkwithtwodelays AT huizhongyang hopfbifurcationanalysisforafourdimensionalrecurrentneuralnetworkwithtwodelays |