Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014 (2014), Article ID 317460, 10 pages
http://dx.doi.org/10.1155/2014/317460
Research Article

A Postverification Method for Solving Forced Duffing Oscillator Problems without Prescribed Periods

1Department of Computer Science and Information Engineering, Fu Jen Catholic University, New Taipei City 24205, Taiwan
2Graduate Institute of Applied Science and Engineering, Fu Jen Catholic University, New Taipei City 24205, Taiwan
3Department of Mathematics, Fu Jen Catholic University, New Taipei City 24205, Taiwan
4General Education Center, St. John’s University, New Taipei City 25135, Taiwan

Received 29 March 2014; Revised 17 July 2014; Accepted 20 July 2014; Published 26 August 2014

Academic Editor: Suh-Yuh Yang

Copyright © 2014 Hong-Yen Lin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. C. J. Luo and J. Huang, “Analytical solutions for asymmetric periodic motions to chaos in a hardening Duffing oscillator,” Nonlinear Dynamics, vol. 72, no. 1-2, pp. 417–438, 2013. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  2. H. Dai, M. Schnoor, and S. N. Atluri, “A simple collocation scheme for obtaining the periodic solutions of the duffing equation, and its equivalence to the high dimensional harmonic balance method: subharmonic oscillations,” Computer Modeling in Engineering and Sciences, vol. 84, no. 5, pp. 459–497, 2012. View at Google Scholar · View at Scopus
  3. N. D. Anh and N. N. Hieu, “The Duffing oscillator under combined periodic and random excitations,” Probabilistic Engineering Mechanics, vol. 30, pp. 27–36, 2012. View at Publisher · View at Google Scholar · View at Scopus
  4. A. Beléndez, M. L. Alvarez, J. Francés et al., “Analytical approximate solutions for the cubic-quintic Duffing oscillator in terms of elementary functions,” Journal of Applied Mathematics, vol. 2012, Article ID 286290, 16 pages, 2012. View at Publisher · View at Google Scholar · View at Scopus
  5. M. Bhatti, M. Lara, and P. Bracken, “Periodic solutions of the Duffing equation,” International Journal of Mathematical Analysis, vol. 2, no. 8, pp. 365–372, 2008. View at Google Scholar · View at MathSciNet
  6. V. Ravichandran, V. Chinnathambi, and S. Rajasekar, “Effect of various periodic forces on Duffing oscillator,” Pramana-Journal of Physics, vol. 67, no. 2, pp. 351–356, 2006. View at Publisher · View at Google Scholar · View at Scopus
  7. V. Ravichandran, V. Chinnathambi, and S. Rajasekar, “Bifurcations and chaos in duffing oscillator driven by different periodic forces,” in Proceedings of the 3rd National Conference on Nonlinear System and Dynamics, 2006.
  8. A. Kenfack, “Bifurcation structure of two coupled periodically driven double-well Duffing oscillators,” Chaos, Solitons and Fractals, vol. 15, no. 2, pp. 205–218, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  9. H. Yin, J. Dai, and H. Zhang, “Phase effect of two coupled periodically driven Duffing oscillators,” Physical Review E, vol. 58, no. 5, pp. 5683–5688, 1998. View at Google Scholar · View at Scopus
  10. R. van Dooren and H. Janssen, “Period doubling solutions in the Duffing oscillator: a Galerkin approach,” Journal of Computational Physics, vol. 82, no. 1, pp. 161–171, 1989. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. Y. Liu and W. Ge, “Positive periodic solutions of nonlinear Duffing equations with delay and variable coefficients,” Tamsui Oxford Journal of Mathematical Sciences, vol. 20, no. 2, pp. 235–255, 2004. View at Google Scholar · View at MathSciNet
  12. M. Kubíček and M. Holodniok, “Codimension-m bifurcation theorems applicable to the numerical verification methods,” Journal of Computational Physics, vol. 70, pp. 203–217, 1987. View at Publisher · View at Google Scholar
  13. J. K. Hale and H. Kocek, Dynamics and Bifurcations, Springer, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  14. A. K. Mallik, “Response of a Hard Duffing Oscillator to Harmonic Excitation—An Overview,” Indian Institute of Technology, Kharagpur, India, 2005.
  15. S. Wiggins, Application to the Dynamics of the Damped, Forced Duffing Oscillator, Springer, New York, NY, USA, 1990.
  16. D. Popa and P. Stan, “Random vibrations of Duffing oscillator with nonlinear elastic characteristic of exponential type,” in Proceedings of the International Conference of Mechanical Engineering, vol. 1, pp. 487–494, University of Craiova, 2010.
  17. A. Tamaevicius, S. Bumeliene, R. Kirvaitis et al., “Autonomous duffing-holmes type chaotic oscillator,” Elektronika ir Elektrotechnika, vol. 93, no. 5, pp. 43–46, 2009. View at Google Scholar
  18. C. Li, E. Wang, and J. Wang, “Potential flux landscapes determine the global stability of a Lorenz chaotic attractor under intrinsic fluctuations,” Journal of Chemical Physics, vol. 136, Article ID 194108, 2012. View at Publisher · View at Google Scholar · View at Scopus
  19. M. Kubíček and M. Marek, Computational Methods in Bifurcation Theory and Dissipative Structures, Springer, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  20. M. Basso, R. Genesio, and A. Tesi, “Stabilizing periodic orbits of forced systems via generalized pyragas controllers,” IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 44, no. 10, pp. 1023–1027, 1997. View at Publisher · View at Google Scholar · View at Scopus
  21. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, 1955. View at MathSciNet
  22. H. Y. Lin, K. C. Jen, and K. C. Jea, “Numerical investigation for bifurcation problems of some nonlinear mathematical models and periodic-doubling bifurcation problem in nonlinear system differential equations,” in Proceedings of the 3rd International Conference of Applied Mathematics, Plovidv, Bulgaria, 2006.