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Mathematical Problems in Engineering
Volume 2017, Article ID 3769870, 8 pages
https://doi.org/10.1155/2017/3769870
Research Article

Chaotic Motion in Forced Duffing System Subject to Linear and Nonlinear Damping

Department of Construction Engineering, National Kaohsiung First University of Science and Technology, Kaohsiung, Taiwan

Correspondence should be addressed to Tai-Ping Chang; wt.ude.tsufkn.smcc@gnahcpt

Received 6 December 2016; Accepted 16 January 2017; Published 31 January 2017

Academic Editor: Jonathan N. Blakely

Copyright © 2017 Tai-Ping Chang. 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. Y.-H. Kao and C.-S. Wang, “Analog study of bifurcation structures in a Van der Pol oscillator with a nonlinear restoring force,” Physical Review E, vol. 48, no. 4, pp. 2514–2520, 1993. View at Publisher · View at Google Scholar · View at Scopus
  2. A. Venkatesan and M. Lakshmanan, “Bifurcation and chaos in the double-well Duffing-van der Pol oscillator: numerical and analytical studies,” Physical Review E. Statistical, Nonlinear, and Soft Matter Physics, vol. 56, no. 6, pp. 6321–6330, 1997. View at Publisher · View at Google Scholar · View at MathSciNet
  3. T. Ueda and H. Kawakami, “Unstable saddle-node connecting orbits in the averaged DuffingRayleigh equation,” ISCAS, vol. 3, pp. 288–291, 1996. View at Google Scholar
  4. F. C. Moon, Chaotic and fractal dynamics, A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York, 1992. View at Publisher · View at Google Scholar · View at MathSciNet
  5. Y.-Z. Wang, F.-M. Li, and Y.-S. Wang, “Influences of active control on elastic wave propagation in a weakly nonlinear phononic crystal with a monoatomic lattice chain,” International Journal of Mechanical Sciences, vol. 106, pp. 357–362, 2016. View at Publisher · View at Google Scholar · View at Scopus
  6. Y.-Z. Wang and F.-M. Li, “Nonlinear primary resonance of nano beam with axial initial load by nonlocal continuum theory,” International Journal of Non-Linear Mechanics, vol. 61, pp. 74–79, 2014. View at Publisher · View at Google Scholar · View at Scopus
  7. Y.-Z. Wang and F.-M. Li, “Nonlinear free vibration of nanotube with small scale effects embedded in viscous matrix,” Mechanics Research Communications, vol. 60, pp. 45–51, 2014. View at Publisher · View at Google Scholar · View at Scopus
  8. R. V. Dooren and H. Janssen, “A continuation algorithm for discovering new chaotic motions in forced Duffing systems,” Journal of Computational and Applied Mathematics, vol. 66, no. 1-2, pp. 527–541, 1996. View at Publisher · View at Google Scholar
  9. X. Han and Q. Bi, “Bursting oscillations in Duffing's equation with slowly changing external forcing,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 10, pp. 4146–4152, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. M. U. Akhmet and M. O. Fen, “Chaotic period-doubling and OGY control for the forced Duffing equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 4, pp. 1929–1946, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. A. K. Mallik, Principles of Vibrations Control, Affiliated East-west Press Private Limited, New Delhi, India, 1990.
  12. B. Ravindra and A. K. Mallik, “Role of nonlinear dissipation in soft Duffing oscillators,” Physical Review E, vol. 49, no. 6, pp. 4950–4954, 1994. View at Publisher · View at Google Scholar · View at Scopus
  13. M. A. F. Sanjuán, “The effect of nonlinear damping on the universal escape oscillator,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 9, no. 4, pp. 735–744, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  14. J. L. Trueba, J. Rams, and M. A. F. Sanjuan, “Analytical estimates of the effect of nonlinear damping in some nonlinear oscillators,” International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, vol. 10, no. 9, pp. 2257–2267, 2000. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. J. P. Baltanás, J. L. Trueba, and M. A. F. Sanjuán, “Energy dissipation in a nonlinearly damped Duffing oscillator,” Physica D: Nonlinear Phenomena, vol. 159, no. 1-2, pp. 22–34, 2001. View at Publisher · View at Google Scholar · View at Scopus
  16. M. Borowiec, G. Litak, and A. Syta, “Vibration of the Duffing oscillator: effect of fractional damping,” Shock and Vibration, vol. 14, no. 1, pp. 29–36, 2007. View at Publisher · View at Google Scholar · View at Scopus
  17. M. S. Siewe, H. Cao, and M. A. F. Sanjuán, “Effect of nonlinear dissipation on the basin boundaries of a driven two-well Rayleigh–Duffing oscillator,” Chaos, Solitons and Fractals, vol. 39, no. 3, pp. 1092–1099, 2009. View at Publisher · View at Google Scholar · View at Scopus
  18. G. Litak, M. Borowiec, A. Syta, and K. Szabelski, “Transition to chaos in the self-excited system with a cubic double well potential and parametric forcing,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2414–2429, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  19. M. S. Siewe, H. Cao, and M. A. F. Sanjuán, “On the occurrence of chaos in a parametrically driven extended Rayleigh oscillator with three-well potential,” Chaos, Solitons and Fractals, vol. 41, no. 2, pp. 772–782, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  20. M. S. Siewe, C. Tchawoua, and P. Woafo, “Melnikov chaos in a periodically driven Rayleigh-Duffing oscillator,” Mechanics Research Communications, vol. 37, no. 4, pp. 363–368, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  21. A. Sharma, V. Patidar, G. Purohit, and K. K. Sud, “Effects on the bifurcation and chaos in forced Duffing oscillator due to nonlinear damping,” Communications in Nonlinear Science and Numerical Simulation, vol. 17, no. 6, pp. 2254–2269, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. Y.-B. Liu, Y.-S. Chen, and Q.-J. Cao, “Analysis method of chaos and sub-harmonic resonance of nonlinear system without small parameters,” Applied Mathematics and Mechanics, vol. 32, no. 1, pp. 1–10, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  23. V. K. Melnikov, “On the stability of the center for time periodic perturbations,” Transactions of the Moscow Mathematical Society, vol. 12, pp. 1–57, 1963. View at Google Scholar
  24. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42, Springer, New York, NY, USA, 1983. View at Publisher · View at Google Scholar · View at MathSciNet
  25. A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, “Determining Lyapunov exponents from a time series,” Physica D: Nonlinear Phenomena, vol. 16, no. 3, pp. 285–317, 1985. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  26. Y.-Z. Wang and F.-M. Li, “Dynamical properties of Duffing–van der Pol oscillator subject to both external and parametric excitations with time delayed feedback control,” Journal of Vibration and Control, vol. 21, no. 2, pp. 371–387, 2015. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  27. K. Yagasaki, “A simple feedback control system: bifurcations of periodic orbits and chaos,” Nonlinear Dynamics, vol. 9, no. 4, pp. 391–417, 1996. View at Publisher · View at Google Scholar · View at Scopus