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Chin. Phys. B, 2015, Vol. 24(12): 120401    DOI: 10.1088/1674-1056/24/12/120401
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Unstable and exact periodic solutions of three-particles time-dependent FPU chains

Liu Qi-Huaia b, Xing Ming-Yana, Li Xin-Xiangc, Wang Chaod
a School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541002, China;
b Guangxi Experiment Center of Information Science, Guilin 541001, China;
c College of Sciences, Shanghai University, Shanghai 200444, China;
d School of Mathematic Sciences, Yancheng Teacher's University, Yancheng 224002, China
Abstract  For lower dimensional Fermi-Pasta-Ulam (FPU) chains, the α-chain is completely integrable and the Hamiltonian of the β-chain can be identified with the Hénon-Heiles Hamiltonian. When the strengths α, β of the nonlinearities depend on time periodically with the same frequencies as the natural angular frequencies, the resonance phenomenon is inevitable. In this paper, for certain periodic functions α(t) and β(t) with resonance frequencies, we give the existence and stability of some nontrivial exact periodic solutions for a one-dimensional α β-FPU model composed of three particles with periodic boundary conditions.
Keywords:  periodic solution      stability      method of averaging  
Received:  20 April 2015      Revised:  18 August 2015      Published:  05 December 2015
PACS:  04.20.Jb (Exact solutions)  
  02.60.-x (Numerical approximation and analysis)  
  02.30.Hq (Ordinary differential equations)  
Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 11301106, 11201288, and 11261013), the Natural Science Foundation of Guangxi Zhuang Autonomous Region, China (Grant No. 2014GXNSFBA118017), the Innovation Project of Graduate Education of Guangxi Zhuang Autonomous Region, China, (Grant No. YCSZ2014143), and the Guangxi Experiment Center of Information Science (Grant No. YB1410).
Corresponding Authors:  Xing Ming-Yan     E-mail:  xinxiang.lee@t.shu.edu.cn

Cite this article: 

Liu Qi-Huai, Xing Ming-Yan, Li Xin-Xiang, Wang Chao Unstable and exact periodic solutions of three-particles time-dependent FPU chains 2015 Chin. Phys. B 24 120401

[1] Fermi E, Pasta J and Ulam S 1955 Los Alamos Report LA-1940 977
[2] Hu X G and Tang Y 2008 Chin. Phys. B 17 426805
[3] Hou Q W and Cao B Y 2012 Chin. Phys. B 21 014401
[4] Giardiná C and Livi R 1998 J. Stat. Phys. 91 1027
[5] Rink B 2001 Commun. Math. Phys. 218 665
[6] Rink B and Verhulst F 2000 Physica A 285 467
[7] Benettin G, Christodoulidi H and Ponno A 2013 J. Stat. Phys. 152 195
[8] Pettini M, Casetti L, Cerruti-Sola M, Franzosi R, and Cohen E 2015 Chaos 15 015106
[9] Antonopoulos C and Bountis T 2006 Phys. Rev. E 73 056206
[10] Bountis T and Skokos H 2012 FPU recurrences and the transition from weak to strong chaos, in: Complex Hamiltonian Dynamics (Berlin/Heidelberg: Springer) pp. 133-134
[11] Penati T, Flach S 2007 Chaos 17 023102
[12] Flach S, Ivanchenko M V and Kanakov O I 2008 Am. J. Phys. 76 453
[13] Xu Q and Tian Q 2013 Chin. Phys. B 22 086302
[14] Sasa S and Komatsu T S 1999 Phys. Rev. Lett. 82 912
[15] Torres P J 2000 Z. Angew. Math. Phys. 51 333
[16] Torres P J 2001 Z. Angew. Math. Phys. 52 535
[17] Sun J and Ma S 2014 J. Math. Anal. Appl. 417 622
[18] Diblík J, FEĆKAN M and Pospíšil M 2013 Miskolc Math. Notes 14 63
[19] Gendelman O 2013 Phys. Rev. E 87 062911
[20] Zhou Q, Lv B B, and Tian Q 2009 Acta Phys. Sin. 58 411 (in Chinese)
[21] Yuan Z Q, Zhu M, and Zheng Z G 2013 Acta Phys. Sin. 62 080504 (in Chinese)
[22] Jia L L, Liu Q H, and Ma Z J 2014 Commun. Nonlinear Sci. Numer. Simul. 19 2715
[23] Ellison J A and Shih H 1995 “The method of averaging in beam dynamics”, in: Accelerator Physics at the Superconducting Super Collider (AIP Publishing) p. 590
[24] Bogoliubov N and Mitropol'skiî I 1961 Asymptotic Methods in the Theory of Nonlinear Oscillations (New York: Gordon and Breach) pp. 387-412
[25] Sanders J A, Verhulst F, and Murdock J A 2007 Averaging Methods in Nonlinear Dynamical Systems (New York: Springer) pp. 33-50
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