Nonlinear Dynamics

, Volume 72, Issue 1–2, pp 389–398 | Cite as

Chaos in an embedded single-walled carbon nanotube

  • Weipeng HuEmail author
  • Zichen Deng
  • Bo Wang
  • Huajiang Ouyang
Original Paper


Considering the geometrical nonlinearity of an embedded single-walled carbon nanotube, the analytical condition and the numerical results of chaotic vibration of the carbon nanotube are presented in this paper. Firstly, based on the Galerkin approximation method, a Duffing-type model is derived from the equation of motion that describes the oscillation of the embedded single-walled carbon nanotube clamped at both ends under a transverse load. And then, the Melnikov function of the Duffing-type model is derived. From the Melnikov function, the analytical condition of the chaos in the nanotube is obtained. Finally, a structure-preserving difference scheme for the original oscillating model is constructed based on the generalized multi-symplectic framework and the chaotic vibration of the nanotube is reproduced to verify the accuracy and the validity of the analytical condition. The analytical condition obtained in this paper gives some guidance on the property studying and the structure designing of some carbon nanotube devices.


Embedded single-walled carbon nanotube Chaos Generalized multi-symplectic Galerkin approximation Melnikov function Structure-preserving 



The authors wish to thank Professor Thomas J. Bridges of Surrey University for giving us several good suggestions. The research is supported by the National Natural Science Foundation of China (11002115, 10972182, 11172239), the Science Foundation of Aviation of China (2010ZB53021), 111 project (B07050) to the Northwestern Polytechnical University, the NPU Foundation for Fundamental Research (JC20110259), the Doctoral Program Foundation of Education Ministry of China (20126102110023), the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21), and the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802).


  1. 1.
    Iijima, S.: Helical microtubules of graphitic carbon. Nature 354(6348), 56–58 (1991) CrossRefGoogle Scholar
  2. 2.
    Tans, S.J., Verschueren, A.R.M., Dekker, C.: Room-temperature transistor based on a single carbon nanotube. Nature 393(6680), 49–52 (1998) CrossRefGoogle Scholar
  3. 3.
    Martel, R., Schmidt, T., Shea, H.R., Hertel, T., Avouris, P.: Single- and multi-wall carbon nanotube field-effect transistors. Appl. Phys. Lett. 73(17), 2447–2449 (1998) CrossRefGoogle Scholar
  4. 4.
    De Heer, W.A., Bacsa, W.S., Châtelain, A., Gerfin, T., Humphrey-Baker, R., Forro, L., Ugarte, D.: Aligned carbon nanotube films: production and optical and electronic properties. Science 268(5212), 845–847 (1995) CrossRefGoogle Scholar
  5. 5.
    Zettl, A.: Extreme oxygen sensitivity of electronic properties of carbon nanotubes. Science 287(5459), 1801–1804 (2000) CrossRefGoogle Scholar
  6. 6.
    Kong, J., Franklin, N.R., Zhou, C., Chapline, M.G., Peng, S., Cho, K., Dai, H.: Nanotube molecular wires as chemical sensors. Science 287(5453), 622–625 (2000) CrossRefGoogle Scholar
  7. 7.
    Che, G., Lakshmi, B.B., Fisher, E.R., Martin, C.R.: Carbon nanotubule membranes for electrochemical energy storage and production. Nature 393(6683), 346–349 (1998) CrossRefGoogle Scholar
  8. 8.
    Coluci, V.R., Legoas, S.B., de Aguiar, M.A.M., Galvao, D.S.: Chaotic signature in the motion of coupled carbon nanotube oscillators. Nanotechnology 16(4), 583–589 (2005). doi: 10.1088/0957-4484/16/4/041 CrossRefGoogle Scholar
  9. 9.
    Wei, D.W., Guo, W.L.: Molecular dynamics simulation of self-assembled carbon nanotubes. Int. J. Nanosci. 5(6), 835–839 (2006) CrossRefGoogle Scholar
  10. 10.
    Conley, W.G., Raman, A., Krousgrill, C.M., Mohammadi, S.: Nonlinear and nonplanar dynamics of suspended nanotube and nanowire resonators. Nano Lett. 8(6), 1590–1595 (2008) CrossRefGoogle Scholar
  11. 11.
    Mayoof, F.N., Hawwa, M.A.: Chaotic behavior of a curved carbon nanotube under harmonic excitation. Chaos Solitons Fractals 42(3), 1860–1867 (2009). doi: 10.1016/j.chaos.2009.03.104 CrossRefGoogle Scholar
  12. 12.
    Hawwa, M.A., Mayoof, F.N.: Nonlinear oscillations of a carbon nanotube resonator. In: ISMA 09, Sharjah – UAE 2009, pp. 1–13 (2009) Google Scholar
  13. 13.
    Hawwa, M.A., Al-Qahtani, H.M.: Nonlinear oscillations of a double-walled carbon nanotube. Comput. Mater. Sci. 48(1), 140–143 (2010) CrossRefGoogle Scholar
  14. 14.
    Wang, L.F., Hu, H.Y., Guo, W.L.: Thermal vibration of carbon nanotubes predicted by beam models and molecular dynamics. Proc. R. Soc. A, Math. Phys. Eng. Sci. 466(2120), 2325–2340 (2010) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Joshi, A.Y., Sharma, S.C., Harsha, S.P.: Chaotic response analysis of single-walled carbon nanotube due to surface deviations. Nano 7(2) (2012). doi: 10.1142/s1793292012500087
  16. 16.
    Lanir, Y., Fung, Y.C.B.: Fiber composite columns under compression. J. Compos. Mater. 6, 387 (1972) Google Scholar
  17. 17.
    Xu, K.Y., Guo, X.N., Ru, C.Q.: Vibration of a double-walled carbon nanotube aroused by nonlinear intertube van der Waals forces. J. Appl. Phys. 99(6), 064303 (2006) CrossRefGoogle Scholar
  18. 18.
    Holmes, P.J., Marsden, J.E.: A partial differential equation with infinitely many periodic orbits: chaotic oscillations of a forced beam. Arch. Ration. Mech. Anal. 76(2), 135–166 (1981) MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Holmes, P.J.: A nonlinear oscillator with a strange attractor. Philos. Trans. R. Soc., Math. Phys. Eng. Sci. 292(1394), 419–448 (1979) zbMATHCrossRefGoogle Scholar
  20. 20.
    Bridges, T.J.: A geometric formulation of the conservation of wave action and its implications for signature and the classification of instabilities. In: Proceeding of the Royal Society London, pp. 1365–1395 (1997) Google Scholar
  21. 21.
    Bridges, T.J.: Multi-symplectic structures and wave propagation. Math. Proc. Camb. Philos. Soc. 121(1), 147–190 (1997) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Hu, W.P., Deng, Z.C., Han, S.M., Zhang, W.R.: Generalized multi-symplectic integrators for a class of Hamiltonian nonlinear wave PDEs. J. Comput. Phys. (2012). doi: 10.1016/ Google Scholar
  23. 23.
    Hu, W.P., Deng, Z.C., Han, S.M., Fan, W.: An implicit difference scheme focusing on the local conservation properties for Burgers equation. Int. J. Comput. Methods 9(2), 1240028 (2012) MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  • Weipeng Hu
    • 1
    • 2
    Email author
  • Zichen Deng
    • 1
    • 3
  • Bo Wang
    • 1
  • Huajiang Ouyang
    • 4
  1. 1.School of Mechanics, Civil Engineering and ArchitectureNorthwestern Polytechnical UniversityXi’anP.R. China
  2. 2.State Key Laboratory of Mechanical System & VibrationShanghai Jiao Tong UniversityShanghaiP.R. China
  3. 3.State Key Laboratory of Structural Analysis of Industrial EquipmentDalian University of TechnologyDalianP. R. China
  4. 4.School of EngineeringUniversity of LiverpoolLiverpoolUK

Personalised recommendations