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Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing
Jiang, Tao1; Yang, Zhiyan1; Jing, Zhujun2
2017-07-01
Source PublicationINTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
ISSN0218-1274
Volume27Issue:8Pages:31
AbstractWe study the Duffing equation with parametric excitation and single external forcing and obtain abundant dynamical behaviors of bifurcations and chaos. The criteria of chaos of the Duffing equation under periodic perturbation are obtained through the Melnikov method. And the existence of chaos of the averaged system of the Duffing equation under the quasi-periodic perturbation Omega = n omega + epsilon nu (where nu is not rational relative to omega) and n = 1, 2, 4, 6 is shown, but the existence of chaos of averaged system of the Duffing equation cannot be proved when n = 3, 5, 7- 15, whereas the occurrence of chaos in the original system can be shown by numerical simulation. Numerical simulations not only show the correctness of the theoretical analysis but also exhibit some new complex dynamical behaviors, including homoclinic or heteroclinic bifurcation surfaces, bifurcation diagrams, Lyapunov exponent diagrams, phase portraits and Poincare maps. We find a large chaotic region with some solitary period parameter points, a large period and quasi-period region with some solitary chaotic parameter points, period-doubling to chaos and chaos to inverse period-doubling, nondense curvilinear chaotic attractor, nonattracting chaotic motion, nonchaotic attracting set, fragmental chaotic attractors. Almost chaotic motion and almost nonchaotic motion appear through adjusting the parameters of the Duffing system, which can be taken as a strategy of chaotic control or a strategy of chaotic motion to nonchaotic motion.
KeywordThe Duffing equation the Melnikov method the second order averaging method bifurcations chaotic attractors
DOI10.1142/S0218127417501255
Language英语
Funding ProjectNational Science Foundations of China[10671063] ; National Science Foundations of China[10801135] ; National Science Foundations of China[61571052]
WOS Research AreaMathematics ; Science & Technology - Other Topics
WOS SubjectMathematics, Interdisciplinary Applications ; Multidisciplinary Sciences
WOS IDWOS:000407487600014
PublisherWORLD SCIENTIFIC PUBL CO PTE LTD
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/26258
Collection中国科学院数学与系统科学研究院
Corresponding AuthorJiang, Tao
Affiliation1.Beijing Wuzi Univ, Sch Informat, Beijing 101149, Peoples R China
2.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
Recommended Citation
GB/T 7714
Jiang, Tao,Yang, Zhiyan,Jing, Zhujun. Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing[J]. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS,2017,27(8):31.
APA Jiang, Tao,Yang, Zhiyan,&Jing, Zhujun.(2017).Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing.INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS,27(8),31.
MLA Jiang, Tao,et al."Bifurcations and Chaos in the Duffing Equation with Parametric Excitation and Single External Forcing".INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 27.8(2017):31.
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