Table of Contents Author Guidelines Submit a Manuscript
Shock and Vibration
Volume 2014, Article ID 131637, 5 pages
http://dx.doi.org/10.1155/2014/131637
Research Article

Chaotic Motions of the Duffing-Van der Pol Oscillator with External and Parametric Excitations

Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing 210016, China

Received 14 May 2014; Revised 7 July 2014; Accepted 7 July 2014; Published 15 July 2014

Academic Editor: Mickaël Lallart

Copyright © 2014 Liangqiang Zhou and Fangqi Chen. 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. L. Ravisankar, V. Ravichandran, and V. Chinnathambi, “Prediction of horseshoe chaos in Duffing-Van der Pol oscillator driven by different periodic forces,” International Journal of Engineering and Science, vol. 1, no. 5, pp. 17–25, 2012. View at Google Scholar
  2. Z. Jing, Z. Yang, and T. Jiang, “Complex dynamics in Duffing-van der Pol equation,” Chaos, Solitons and Fractals, vol. 27, no. 3, pp. 722–747, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. Z. Qin and Y. Chen, “Singularity analysis of Duffing-van der Pol system with two bifurcation parameters under multi-frequency excitations,” Applied Mathematics and Mechanics: English Edition, vol. 31, no. 8, pp. 1019–1026, 2010. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  4. A. T. Y. Leung, H. X. Yang, and P. Zhu, “Periodic bifurcation of Duffing-van der Pol oscillators having fractional derivatives and time delay,” Communications in Nonlinear Science and Numerical Simulation, vol. 19, no. 40, pp. 1013–1018, 1993. View at Google Scholar
  5. E. F. Udwadia and E. Cho, “First integrals and solutions of Duffing-Van der Pol type equations,” Journal of Applied Mechanics-Transactions of the ASME, vol. 81, no. 3, Article ID 034501, 2014. View at Google Scholar
  6. A. T. Y. Leung, H. X. Yang, and P. Zhu, “Neimark bifurcations of a generalized Duffing-Van der Pol oscillator with nonlinear fractional order damping,” International Journal of Bifurcation and Chaos, vol. 23, no. 11, Article ID 1350177, 19 pages, 2013. View at Publisher · View at Google Scholar
  7. Z. M. Ge and S. Y. Li, “Chaos generalized synchronization of new Mathieu-VAN der POL systems with new Duffing-VAN der POL systems as functional system by {GYC} partial region stability theory,” Applied Mathematical Modelling, vol. 35, no. 11, pp. 5245–5264, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. S. G. Zhang, Z. Y. Zhu, and Z. Guo, “Primary resonance and bifurcations in damped and driven Duffing-Van der Pol system,” Advanced Materials Research, vol. 216, no. 1-2, pp. 782–786, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. X. Li, H. Zhang, and L. Zhang, “Response of the Duffing-van der Pol oscillator under position feedback control with two time delays,” Shock and Vibration, vol. 18, no. 1-2, pp. 377–386, 2011. View at Publisher · View at Google Scholar · View at Scopus
  10. Y. X. Shi, D. Y. Bai, and W. J. Tao, “Chaos and control of the Duffing-van der Pol equation with two external periodic excitations,” Journal of Hebei Normal University, vol. 34, no. 6, pp. 631–635, 2010. View at Google Scholar
  11. J. Yu, W. Pan, and R. Zhang, “Period-doubling cascades and strange attractors in extended Duffing-Van der Pol oscillator,” Communications in Theoretical Physics, vol. 51, no. 5, pp. 865–868, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. J. Yu and J. R. Li, “Investigation on dynamics of the extended Duffing-Van der Pol system,” Zeitschrift fur Naturforschung A: A Journal of Physical Sciences, vol. 64, no. 5-6, pp. 341–346, 2009. View at Google Scholar · View at Scopus
  13. J. Yu, Z. Xie, and L. Yu, “Complex dynamics in a Duffing-Van der Pol oscillator with ψ6 potential,” Journal of the Physical Society of Japan, vol. 77, no. 11, Article ID 114003, 2008. View at Publisher · View at Google Scholar · View at Scopus
  14. H. G. Patel and S. N. Sharma, “Filtering for a Duffing-van der Pol stochastic differential equation,” Applied Mathematics and Computation, vol. 226, pp. 386–397, 2014. View at Publisher · View at Google Scholar · View at MathSciNet
  15. C. Li, W. Xu, L. Wang, and D. X. Li, “Stochastic responses of Duffing-Van der Pol vibro-impact system under additive colored noise excitation,” Chinese Physics B, vol. 22, no. 11, Article ID 110205, 2013. View at Publisher · View at Google Scholar
  16. J. B. Li, Chaos and Melnikov Method, Chongqing University Press, Chongqing, China, 1989, (Chinese).
  17. S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos, vol. 2 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  18. M. S. Siewe, C. Tchawoua, and S. Rajasekar, “Homoclinic bifurcation and chaos in ϕ6-Rayleigh oscillator with three wells driven by an amplitude modulated force,” International Journal of Bifurcation and Chaos, vol. 21, no. 6, pp. 1583–1593, 2011. View at Publisher · View at Google Scholar · View at Scopus
  19. V. Ravichandran, V. Chinnathambi, and S. Rajasekar, “Homoclinic bifurcation and chaos in Duffing oscillator driven by an amplitude-modulated force,” Physica A: Statistical Mechanics and its Applications, vol. 376, pp. 223–236, 2007. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. C. Castilho and M. Marchesin, “A practical use of the Melnikov homoclinic method,” Journal of Mathematical Physics, vol. 50, no. 11, Article ID 112704, 11 pages, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  21. L. Shi, Y. Zou, and T. Küpper, “Melnikov method and detection of chaos for non-smooth systems,” Acta Mathematicae Applicatae Sinica: English Series, vol. 29, no. 4, pp. 881–896, 2013. View at Publisher · View at Google Scholar · View at MathSciNet