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Complex dynamics in pendulum equation with parametric and external excitations I
Jing, Zhujun; Yang, Jianping
2006-10-01
Source PublicationINTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS
ISSN0218-1274
Volume16Issue:10Pages:2887-2902
AbstractPendulum equation with parametric and external excitations is investigated in (1) and (II). In (1), by applying Melnikov's method, we prove the criterion of existence of chaos under periodic perturbation. The numerical simulations, including bifurcation diagram of fixed points, bifurcation diagram of system in three- and two-dimensional space, homoclinic and heteroclinic bifurcation surface, Maximum Lyapunov exponent, phase portraits, Poincare map, are plotted to illustrate theoretical analysis, and to expose the complex dynamical behaviors including the period-n (n = 2 to 6, 10, 15 and 20) orbits in different chaotic regions, interlocking periodic orbits, symmetry-breaking of periodic orbit, cascade of period-doubling bifurcations from period-5 and -10 orbits, reverse period-doubling bifurcation, onset of chaos which occurs more than once for a given external frequency or parametric frequency and chaos suddenly converting to periodic orbits, sudden jump in the size of attractors which is associated with the transverse intersection of stable and unstable manifolds of perturbed saddle, hopping behavior of chaos, transient chaos with complex periodic windows and interior crisis, varied chaotic attractors including the more than three-band and eight-band chaotic attractors, chaotic attractor after strange nonchaotic attractor. In particular, we observe that the system can leave chaotic region to periodic motion by adjusting damping delta, spring constant alpha and frequency Omega of parametric excitation which can be considered as a control strategy. In (11), we will investigate the complex dynamics under quasi-periodic perturbation.
Keywordpendulum equation Melnikov's method bifurcations chaos
Language英语
WOS Research AreaMathematics ; Science & Technology - Other Topics
WOS SubjectMathematics, Interdisciplinary Applications ; Multidisciplinary Sciences
WOS IDWOS:000243320500007
PublisherWORLD SCIENTIFIC PUBL CO PTE LTD
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Document Type期刊论文
Identifierhttp://ir.amss.ac.cn/handle/2S8OKBNM/3921
Collection中国科学院数学与系统科学研究院
Corresponding AuthorJing, Zhujun
Affiliation1.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100080, Peoples R China
2.Hunan Normal Univ, Dept Math, Changsha 410081, Peoples R China
3.Chinese Acad Sci, Grad Sch, Beijing 100039, Peoples R China
Recommended Citation
GB/T 7714
Jing, Zhujun,Yang, Jianping. Complex dynamics in pendulum equation with parametric and external excitations I[J]. INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS,2006,16(10):2887-2902.
APA Jing, Zhujun,&Yang, Jianping.(2006).Complex dynamics in pendulum equation with parametric and external excitations I.INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS,16(10),2887-2902.
MLA Jing, Zhujun,et al."Complex dynamics in pendulum equation with parametric and external excitations I".INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS 16.10(2006):2887-2902.
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