Complex dynamics in physical pendulum equation with suspension axis vibrations

doi:10.1007/s10255-008-8276-6

CSpace

	Complex dynamics in physical pendulum equation with suspension axis vibrations
	Fu, Xiang-ling 1,2; Deng, Jin 3; Jing, Zhu-jun 1,4
	2010
发表期刊	ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES
ISSN	0168-9673
卷号	26 期号:1 页码:55-78
摘要	The physical pendulum equation with suspension axis vibrations is investigated. By using Melnikov's method, we prove the conditions for the existence of chaos under periodic perturbations. By using second-order averaging method and Melinikov's method, we give the conditions for the existence of chaos in an averaged system under quasi-periodic perturbations for Omega = n omega + E >nu, n = 1 - 4, where nu is not rational to omega. We are not able to prove the existence of chaos for n = 5 - 15, but show the chaotic behavior for n = 5 by numerical simulation. By numerical simulation we check on our theoretical analysis and further exhibit the complex dynamical behavior, including the bifurcation and reverse bifurcation from period-one to period-two orbits; the onset of chaos, the entire chaotic region without periodic windows, chaotic regions with complex periodic windows or with complex quasi-periodic windows; chaotic behaviors suddenly disappearing, or converting to period-one orbit which means that the system can be stabilized to periodic motion by adjusting bifurcation parameters alpha, delta, f (0) and Omega; and the onset of invariant torus or quasi-periodic behaviors, the entire invariant torus region or quasi-periodic region without periodic window, quasi-periodic behaviors or invariant torus behaviors suddenly disappearing or converting to periodic orbit; and the jumping behaviors which including from periodone orbit to anther period-one orbit, from quasi-periodic set to another quasi-periodic set; and the interleaving occurrence of chaotic behaviors and invariant torus behaviors or quasi-periodic behaviors; and the interior crisis; and the symmetry breaking of period-one orbit; and the different nice chaotic attractors. However, we haven't find the cascades of period-doubling bifurcations under the quasi-periodic perturbations and show the differences of dynamical behaviors and technics of research between the periodic perturbations and quasi-periodic perturbations.
关键词	Pendulum equation suspension axis vibrations averaging method Melnikov's method bifurcations chaos
DOI	10.1007/s10255-008-8276-6
语种	英语
资助项目	National Natural Science Foundation of China[10671063]
WOS研究方向	Mathematics
WOS类目	Mathematics, Applied
WOS记录号	WOS:000272628300006
出版者	SPRINGER HEIDELBERG
引用统计
文献类型	期刊论文
条目标识符	http://ir.amss.ac.cn/handle/2S8OKBNM/10334
专题	中国科学院数学与系统科学研究院
通讯作者	Jing, Zhu-jun
作者单位	1.Hunan Normal Univ, Coll Math & Comp Sci, Changsha 410081, Hunan, Peoples R China 2.Hunan Univ Sci & Technol, Sch Math, Xiangtan 411201, Peoples R China 3.Hunan Inst Engn, Dept Math & Phys, Xiangtan 411104, Peoples R China 4.Chinese Acad Sci, Acad Math & Syst Sci, Beijing 100190, Peoples R China
推荐引用方式 GB/T 7714	Fu, Xiang-ling,Deng, Jin,Jing, Zhu-jun. Complex dynamics in physical pendulum equation with suspension axis vibrations[J]. ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES,2010,26(1):55-78.
APA	Fu, Xiang-ling,Deng, Jin,&Jing, Zhu-jun.(2010).Complex dynamics in physical pendulum equation with suspension axis vibrations.ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES,26(1),55-78.
MLA	Fu, Xiang-ling,et al."Complex dynamics in physical pendulum equation with suspension axis vibrations".ACTA MATHEMATICAE APPLICATAE SINICA-ENGLISH SERIES 26.1(2010):55-78.