Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays
We consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the di...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Wiley
2011-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2011/569141 |
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| _version_ | 1850177281687814144 |
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| author | Xiaobing Zhou Murong Jiang Xiaomei Cai |
| author_facet | Xiaobing Zhou Murong Jiang Xiaomei Cai |
| author_sort | Xiaobing Zhou |
| collection | DOAJ |
| description | We consider the van der Pol equation with discrete and distributed delays. Linear
stability of this equation is investigated by analyzing the transcendental characteristic equation of
its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations
by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf
bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis. |
| format | Article |
| id | doaj-art-c31d4f9731df4e1d89ac39abffacf3d6 |
| institution | OA Journals |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2011-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-c31d4f9731df4e1d89ac39abffacf3d62025-08-20T02:19:01ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2011-01-01201110.1155/2011/569141569141Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed DelaysXiaobing Zhou0Murong Jiang1Xiaomei Cai2School of Information Science and Engineering, Yunnan University, Kunming 650091, ChinaSchool of Information Science and Engineering, Yunnan University, Kunming 650091, ChinaBureau of Asset Management, Yunnan University, Kunming 650091, ChinaWe consider the van der Pol equation with discrete and distributed delays. Linear stability of this equation is investigated by analyzing the transcendental characteristic equation of its linearized equation. It is found that this equation undergoes a sequence of Hopf bifurcations by choosing the discrete time delay as a bifurcation parameter. In addition, the properties of Hopf bifurcation were analyzed in detail by applying the center manifold theorem and the normal form theory. Finally, some numerical simulations are performed to illustrate and verify the theoretical analysis.http://dx.doi.org/10.1155/2011/569141 |
| spellingShingle | Xiaobing Zhou Murong Jiang Xiaomei Cai Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays Discrete Dynamics in Nature and Society |
| title | Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays |
| title_full | Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays |
| title_fullStr | Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays |
| title_full_unstemmed | Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays |
| title_short | Hopf Bifurcation Analysis for the van der Pol Equation with Discrete and Distributed Delays |
| title_sort | hopf bifurcation analysis for the van der pol equation with discrete and distributed delays |
| url | http://dx.doi.org/10.1155/2011/569141 |
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