Chaos and Sub-Harmonic Resonance of Nonlinear System Without Small Parameters

LIU Yan-bin; CHEN Yu-shu; CAO Qing-jie

doi:10.3879/j.issn.1000-0887.2011.01.001

Volume 32 Issue 1

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LIU Yan-bin, CHEN Yu-shu, CAO Qing-jie. Chaos and Sub-Harmonic Resonance of Nonlinear System Without Small Parameters[J]. Applied Mathematics and Mechanics, 2011, 32(1): 1-10. doi: 10.3879/j.issn.1000-0887.2011.01.001

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Chaos and Sub-Harmonic Resonance of Nonlinear System Without Small Parameters

doi: 10.3879/j.issn.1000-0887.2011.01.001

School of Astronautics, Harbin Institute of Technology, China PO Box 137, Harbin 150001, P. R. China

Received Date: 2010-09-15
Rev Recd Date: 2010-12-07
Publish Date: 2011-01-15

Abstract

Abstract

Melnikov method was especially important to detect the presence of transverse homoclinic orbits and occurrence of homoclinic bifurcations.Unfortunately traditional Melnikov methods strongly depend on small parameter,which could not exist in most of the practice physical systems.Those methods were limited in dealing with the system with strongly nonlinear.A procedure to study the chaos and sub-harmonic resonance of strongly nonlinear practice systems by employing homotopy method which was used to extend Melnikov functions to strongly nonlinear systems was presented.Applied to a given example,the procedure shows the efficiencies in the comparison of the theoretical results and numerical simulation.
- homotopy,
- Melnikov function,
- chaos,
- sub-harmonic

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References

[1]	CHEN Yu-shu, Leung Andrew Y T. Bifurcation and Chaos in Engineering[M]. New York: Springer ,1998.
[2]	Wiggins S.Introduction to Applied Nonlinear Dynamical Systems and Chaos[M].New York:Springer-Verlag,1990.
[3]	Greenspan B D, Holmes P J.Homoclinic orbits, subharmonics and global bifurcations in forced oscillations[C] Barenblatt G, Iooss G, Joseph D D. Nonlinear Dynamics and Turbulence. London: Pitman, 1983: 172-214.
[4]	Wiggins S. Global Bifurcations and Chaos[M].New York:Springer-Verlag,1988.
[5]	Liao S J. Beyond Perturbation：Introduction to Homotopy Analysis Method[M]. Bpca Taton: Chapmaen Hall/CRC Press, 2003.
[6]	Liao S J. The proposed homotopy analysis techniques for the solution of nonlinear problems[D]. Ph D dissertation. Shanghai: Shanghai Jiao Tong University, 1992.
[7]	Liao S J. On the homotopy analysis method for nonlinear problems[J].Appl Math Comput, 2004,147(2): 499-513. doi: 10.1016/S0096-3003(02)00790-7
[8]	Liao S J. A kind of approximate solution technique which does not depend upon small parameters—Ⅱ:an application in fluid mechanics[J]. Int J Non-Linear Mech, 1997, 32(4):815-822. doi: 10.1016/S0020-7462(96)00101-1
[9]	Liao S J. An explicit, totally analytic approximation of Blasius viscous flow problems[J]. Int J Non-Linear Mech, 1999,34(4):759-785 doi: 10.1016/S0020-7462(98)00056-0