Complex dynamics in pendulum equation with parametric and external excitations I
Jing, Zhujun; Yang, Jianping
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
WOS Research AreaMathematics ; Science & Technology - Other Topics
WOS SubjectMathematics, Interdisciplinary Applications ; Multidisciplinary Sciences
WOS IDWOS:000243320500007
Citation statistics
Document Type期刊论文
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.
Files in This Item:
There are no files associated with this item.
Related Services
Recommend this item
Usage statistics
Export to Endnote
Google Scholar
Similar articles in Google Scholar
[Jing, Zhujun]'s Articles
[Yang, Jianping]'s Articles
Baidu academic
Similar articles in Baidu academic
[Jing, Zhujun]'s Articles
[Yang, Jianping]'s Articles
Bing Scholar
Similar articles in Bing Scholar
[Jing, Zhujun]'s Articles
[Yang, Jianping]'s Articles
Terms of Use
No data!
Social Bookmark/Share
All comments (0)
No comment.

Items in the repository are protected by copyright, with all rights reserved, unless otherwise indicated.