Bifurcation Analysis and Chaos Control in a Discrete Epidemic System
The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior R+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation h...
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| Format: | Article |
| Language: | English |
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Wiley
2015-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2015/974868 |
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| author | Wei Tan Jianguo Gao Wenjun Fan |
| author_facet | Wei Tan Jianguo Gao Wenjun Fan |
| author_sort | Wei Tan |
| collection | DOAJ |
| description | The dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior R+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the system x→x+δ[rN(1-N/K)-βxy/N-(μ+m)x], y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method. |
| format | Article |
| id | doaj-art-129976cc574e482abf9532eac83a0d3b |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2015-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-129976cc574e482abf9532eac83a0d3b2025-02-03T01:01:07ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2015-01-01201510.1155/2015/974868974868Bifurcation Analysis and Chaos Control in a Discrete Epidemic SystemWei Tan0Jianguo Gao1Wenjun Fan2Department of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, ChinaDepartment of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, ChinaDepartment of Mathematics and Information Science, Beifang University of Nationalities, Yinchuan, Ningxia 750021, ChinaThe dynamics of discrete SI epidemic model, which has been obtained by the forward Euler scheme, is investigated in detail. By using the center manifold theorem and bifurcation theorem in the interior R+2, the specific conditions for the existence of flip bifurcation and Neimark-Sacker bifurcation have been derived. Numerical simulation not only presents our theoretical analysis but also exhibits rich and complex dynamical behavior existing in the case of the windows of period-1, period-3, period-5, period-6, period-7, period-9, period-11, period-15, period-19, period-23, period-34, period-42, and period-53 orbits. Meanwhile, there appears the cascade of period-doubling 2, 4, 8 bifurcation and chaos sets from the fixed point. These results show the discrete model has more richer dynamics compared with the continuous model. The computations of the largest Lyapunov exponents more than 0 confirm the chaotic behaviors of the system x→x+δ[rN(1-N/K)-βxy/N-(μ+m)x], y→y+δ[βxy/N-(μ+d)y]. Specifically, the chaotic orbits at an unstable fixed point are stabilized by using the feedback control method.http://dx.doi.org/10.1155/2015/974868 |
| spellingShingle | Wei Tan Jianguo Gao Wenjun Fan Bifurcation Analysis and Chaos Control in a Discrete Epidemic System Discrete Dynamics in Nature and Society |
| title | Bifurcation Analysis and Chaos Control in a Discrete Epidemic System |
| title_full | Bifurcation Analysis and Chaos Control in a Discrete Epidemic System |
| title_fullStr | Bifurcation Analysis and Chaos Control in a Discrete Epidemic System |
| title_full_unstemmed | Bifurcation Analysis and Chaos Control in a Discrete Epidemic System |
| title_short | Bifurcation Analysis and Chaos Control in a Discrete Epidemic System |
| title_sort | bifurcation analysis and chaos control in a discrete epidemic system |
| url | http://dx.doi.org/10.1155/2015/974868 |
| work_keys_str_mv | AT weitan bifurcationanalysisandchaoscontrolinadiscreteepidemicsystem AT jianguogao bifurcationanalysisandchaoscontrolinadiscreteepidemicsystem AT wenjunfan bifurcationanalysisandchaoscontrolinadiscreteepidemicsystem |