| Bifurcation of Periodic Orbits and Chaos in Duffing Equation |
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| 作者姓名： | Mei-xiang Cai Jian-ping Yang |
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| 作者单位： | [1]Department of Mathcmatics, Hunan Normal University, Changsha 410081, China [2]Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, Chian, and Graduate School of the Chinese Academy of Sciences, Beijing 100039 |
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| 基金项目： | Supported by the National Natural Science Foundation of China （No.10371037）, and by Chinese Academy Scicnces （KZCX2-SW-118） |
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| 摘 要： | Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated, The conditions of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling bifurcations from period 1, 2, and 3 orbits, period-windows （period-2, 3 and 5） in chaotic regions.
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| 关 键 词： | Melnikov法 二阶平均法 混沌理论 外强制 相移 |
| 收稿时间： | 2005-06-23 |
| 修稿时间： | 2005-06-23 |
Bifurcation of Periodic Orbits and Chaos in Duffing Equation |
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| Mei-xiang Cai Jian-ping Yang.Bifurcation of Periodic Orbits and Chaos in Duffing Equation[J].Acta Mathematicae Applicatae Sinica,2006,22(3):495-508. |
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| Authors: | Mei-xiang Cai Jian-ping Yang |
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| Institution: | (1) Department of Mathematics, Hunan Normal University, Changsha, 410081, China;(2) Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100080, Chian;(3) Graduate School of the Chinese Academy of Sciences, Beijing, 100039, China |
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| Abstract: | Abstract
Duffing equation with fifth nonlinear-restoring force, one external forcing and a phase shift is investigated. The conditions
of existences for primary resonance, second-order, third-order subharmonics, morder subharmonics and chaos are given by using
second-averaging method, Melnikov methods and bifurcation theory. Numerical simulations including bifurcation diagrams, bifurcation
surfaces, phase portraits, not only show the consistence with the theoretical analysis, but also exhibit the new dynamical
behaviors. We show the onset of chaos, chaos suddenly disappearing to period orbit, one-band and double-band chaos, period-doubling
bifurcations from period 1, 2, and 3 orbits, period-windows (period-2, 3 and 5) in chaotic regions.
Supported by the National Natural Science Foundation of China (No.10371037), and by Chinese Academy Sciences (KZCX2-SW-118) |
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| Keywords: | Duffing equation Melnikov's method second-order averaging method chaos |
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